DESIGN AND ANALYSIS OF DISC DRIVE
SERVOMECHANICAL SUBSYSTEM
by
Shirish D. Bahirat
B.E., University of Pune, 1993
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2004
This thesis for the Master of Science
degree by
Shirish D. Bahirat
has been approved
by
Date
Gita Alaghband
Bahirat, Shirish D. (M.S., Electrical Engineering)
Design and Analysis of Disc Drive ServoMechanical Subsystem
Thesis directed by Professor Hamid Fardi
ABSTRACT
For a control engineer it is very fascinating not only to understand the
disc drive control technology, but to also find ways to improve its perfor
mance. Disc drive control technology includes various state of art control
applications and requires continuous improvement over the time. The goal
of this thesis is to design and analyze a digital control system for a disc
drive. Theoretical algorithms and equations are included as well as valida
tion simulations, for the tracking controller. The development part of the
work carried out using various methodologies, in search of the best pos
sible option.This thesis not only includes efforts to understand disc drive
digital control technology, but also extends to searches for new innovative
solutions towards increased performance. Significant amount of knowledge
was gained thorough practical experience as well as from classroom learning
at the University of Colorado, at Denver and Boulder. Analysis and work
presented in this thesis is validated with practical implementation.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Hamid Fardi
m
DEDICATION
I dedicate this thesis to my wife Pooja, son Ameya and daughter Neha for
their unfaltering understanding and support while I was working on this.
ACKNOWLEDGEMENT
Develop clear understanding towards disc drive digital control was one of
the key goals for this thesis work. Apart from learning, efforts were made to
improve and solve performance issues for tracking controller and external
disturbance cancellation. This thesis presents innovative approaches for
feed forward and dual stage inverse control.
Also, I hope this thesis will provide inspiration to the new engineering stu
dents, by introducing them with the work being done in the disc drive indus
try that is based on the things they are learning in the classroom. Similar
digital control technologies are used in various industries such as, aerospace,
communication, robotics, industrial automation, automotive, manufactur
ing and defense.
I would like to acknowledge Dr. Hamid Fardi for providing valuable guid
ance and inputs to this thesis. Classwork with Dr. Miloje Radenkovic facil
itated adaptive algorithm developments. Robust control class work as well
as publications from Dr. Peter Young made the controller synthesis work
feasible and practical. Also I would like to acknowledge inspiration from
Dr. Hai Ho, former director of Seagate Technology Advance Servo Tech
nology Development. The valuable inputs from thesis committee members
Dr. Joseph Hibey, Dr. Titsa Papantoni and Dr. Gita Alaghband helped to
improve the presentation style and technicalities.
CONTENTS
Figures.......................i.......
Tables ............................
Chapter
1. Introduction ..............'........
1.1 Key Components of a Disc Drive .. .
1.1.1 Disc Pack Assembly . ..........
1.1.2 Actuator Assembly................
1.2 Key Components of a Digital Servo
1.2.1 Seek .............. .............
1.2.2 Settle ...................
1.2.3 Tracking ........................
1.2.4 Spin ..........................
2. Model Development ...............
2.1 Obtaining the Generalized Plant .
2.1.1 Single Stage Analytical Plant Model
2.1.2 Dual Stage Analytical Plant Model .
2.1.3 Seek Parametric Plant Model . .
2.1.4 Transconductance Loop Design .
ix
xiii
1
2
3
3
7
7
8
9
11.
14
15
16
18
22
23
vi
2.2 Modelling High Frequency Dynamics........................... . 24
2.3. Estimator Model................................................ 27
3. Velocity Mode Seek Control..................................... 34
3.1 Deriving Seek Control Algorithm............................. 36
3.1.1 Exponential Arrival and Switch Point.......................... 38
3.2 Addition of Back EMF ......................................... 42
4. Track Follow Control ........................................ 51
4.1 General Control Problem Formulation........................... 52
4.1.1 Signal Norms and Their Properties........................... . 54
4.1.2 Obtaining the Generalized Plant ............................ 55
4.1.3 Plant Uncertainty........................................... 59
4.1.4 Dual Stage Servo Controller Synthesis....................... . 62
4.2 Controller Synthesis.......................................... 66
4.2.1 Structured Singular Value fx................................. 69
4.2.2 Control Design via (x Synthesis............................... 75
4.2.3 Synthesis Summary........................................... 79
4.3 Time Domain Simulation........................................ 79
4.3.1 Dual Stage Controller Simulation.............................. 80
4.3.2 Dual Stage Matlab Batch Simulation............................ 82
5. External Disturbance Cancellation ............................. 94
5.1 Generalized Disturbance Cancellation Filter................... 95
5.2 Relaxing the Model Knowledge.................................. 98
vii
5.2.1 Adaptive Algorithm Development ............................ 99
5.2.2 Applying Adaptive Algorithm................................102
5.3 Simulation Example..........................................102
6. Dual Stage Inverse Control ................................ 108
6.1 Disturbance Estimation (j,..................................108
6.2 Microactuator Based Inverse Control........................113
6.3 Inverse Adaptive Control....................................115
7. Repeatable RunOut Cancellation.............................. 125
7.1 Zero Acceleration Path..................................... 126
7.2 Adaptive Feed Forward Compensation .......................128
7.3 Coherent Feed Forward Compensation..........................129
8. Conclusion................................................ 137
8.1 Synopsis ...................................................137
8.2 Challenges in Future Disc Drives............................139
Appendix
A. Nomenclature ............................................ 142
)
References................................................... 145
vm
FIGURES
Figure
1.1 Finite element model of disk drive ....................... 12
1.2 Finite element solution of disk drive . ............. 13
2.1 Single stage plant response ......'....................... 30
2.2 Dual stage plant response ................................ 31
2.3 Block diagram with dual stage with parallel structure .... 31
2.4 Block diagram with dual stage with decoupled structure . 32
2.5 Decoupled plant........................................... 32
2.6 Simplified plant ......................................... 32
2.7 State space model based on FEM............................. 33
3.1 Position mode seek ...................................... 35
3.2 Velocity mode seek.......................................... 47
3.3 Position vs velocity...................................... . 47
3.4 Seek length vs phase plane ................................. 48
3.5 Feed forward term with respect to acceleration ............ 48
3.6 Matlab seek simulation .................................... 49
IX
3.7 Velocity mode seek simulation............................... 49
3.8 Velocity error high at start and low at end................. 50
4.1 General control problem formulation ........................ 85
4.2 Multiplicative plant uncertainty............................ 86
4.3 Additive plant uncertainty.................................. 86
4.4 Parametric uncertainty...................................... 87
4.5 Synthesis weights............................................. 87
4.6 Sensitivity vs gain and phase margin.......................... 88
4.7 Synthesis interconnect ....................................... 88
4.8 Synthesis results ........................................... 89
4.9 //synthesis interconnect .................................... 89
4.10 Controller design by//synthesis............................. 90
4.11 Controller design by ifoosynthesis......................... 90
4.12 Simulating dual stage environment .......................... 91
4.13 Matlab state space implementation............................ 91
4.14 Time domain simulated plant vs modelled plant................ 92
4.15 Simulated vs modelled loop and controller.................... 92
4.16 Simulated nonrepeatable runout............................. 93
5.1 General control configuration ................................ 95
5.2 Simple simulation diagram.....................................103
5.3 Simple simulation results.....................................104
5.4 Control configuration with adaptive feed forward .............104
5.5 RVFF with NLMS simulation.....................................105
x
5.6 RVFF with IIRLMS simulation.................................105
5.7 RVFF with LMS simulation....................................106
5.8 RVFF with RLS simulation .............................. 106
5.9 RVFF with NGD simulation...................................107
5.10 RVFF with VGNLMS simulation.................................107
6.1 Single stage inverse control.................................109
6.2 Inverse control simulation with delta 1 ........119
6.3 Inverse control simulation with delta 3......................119
6.4 Inverse dual stage control ..................................120
6.5 Non repeatable run out with dual stage controller....120
6.6 Non repeatable run out with dual stage inverse controller . 121
6.7 Inverse adaptive dual stage control..........................121
6.8 Micro actuator system identification.........................122
6.9 Stroke requirements with and without inverse control .... 122
6.10 Estimated and modelled disturbance......................... 123
6.11 HR RLS adaption performance.................................123
6.12 RV disturbance rejection....................................124
7.1 Close loop transfer function 126
7.2 Close loop with repeatable run out...........................131
7.3 Repeatable run out without ZAP...............................132
7.4 Repeatable run out with ZAP..................................133
7.5 Adaptive feed forward compensation simulation................134
xi
7.6 PES comparison with and w/o adaptive feed forward com
pensation ....................................................135
7.7 RRO comparison schemes simulation................... 135
7.8 RRO simulation results...................................136
Xll
TABLES
Table
5.1 RVFF simulation result summary
xiu
1. Introduction
Every notebook, desktop computer and server contain one or more disk
drives. Mainframe and supercomputers are normally connected to hun
dreds of them. Modern VCRtype devices, cars and camcorders use hard
disks instead of tapes. Hard disks do one thing very well: they store chang
ing digital information in a relatively permanent form, providing computers
the ability to store very large data banks, even when thepower is turned
off. Hard disk drives were invented in the 1950s'; these started as large
disks up to 20 inches in diameter holding just a few megabytes. They were
originally called fixed disks or Winchesters (name of a popular IBM prod
uct); later they termed as hard disks or disc drives. Hard disks have stiff
spinning platters or discs that are coated with the high permeable magnetic
materials. At the simplest level, a hard disk is not much different from a
cassette recorder and tape. Both hard disks and cassette tapes use mag
netic recording techniques and share the major benefits of magnetic storage:
easily erased and restored data banks, and durability of stored information.
The magnetic recording material on a cassette tape is coated onto a thin
plastic strip. Similarly, the magnetic recording material of a hard disc drive
is layered onto a highprecision aluminum or glass disk. On a hard disk,
1
one can move to any point on the surface of the disk almost instantly, when
seeking a specific data track; the read and write head flies over the disk,
never actually touching it. A harddisk platter can spin underneath its head
at speeds up to 3,000 inches per second comes pounding to about 170 miles
or 272 kilometer per hour. The information on a hard disk is stored in
extremely small magnetic domains and the size of these domains is made
possible by the precision of the platter and the speed of the medium. With
the growth of Internet and the modern consumer applications, such as per
sonal video recorders, the market for disc drives is continuously growing,
demands for increased capacity, functionality and robustness. Digital servo
plays a significant role in the modern high capacity disc drives by accu
rately positioning the read and write head over a data track or by moving
the head to alternate data tracks. Read and write head media subsystems
are another critical part of disc drive and involves complex and advanced
technologies. This thesis focuses on the servo mechanical subsystem; its
design and analysis, for a high capacity disc drive digital control system
design.
1.1 Key Components of a Disc Drive
Hard disc drive mechanics is commonly referred to as head disc assembly or
HDA. It mainly consists of three subassemblies, the disc pack assembly, the
actuator assembly also called the head stack assembly or HSA and the base
assembly. Following is the overview of the various disc drive components,
to mainly solicit the understandings to how such components affect the
2
digital servo controller design requirements. Figure (1.1) shows a disc drive
dynamics finite element analysis model, which will be utilized for control
system modelling design via a state space Matlab model. This kind of
analysis may represent a proactive approach towards better resolution of
any design issues that might limit the control system performance.
1.1.1 Disc Pack Assembly
Disc pack assembly consists of a single or multiple discs mounted on a
spindle motor that spins at constant speed, anywhere from 5400 to 12000
RPM. Magnetic discs are prewritten with servo sectors or servo wages.
Each servo wedge includes 4 servo bursts; demodulating the servo bust
signal can provide position information of the read and write head, with
respect to the servo track center. The difference between head position
and track center is called position error signal. Disc resonance is one of
the many sources of position error in the tracking control system. Fluid or
hydro dynamic bearings, used in the spindle motor, minimize motor induced
excitation.
1.1.2 Actuator Assembly
A rotary Voice Coil Motor(VCM) positions heads over servo or data tracks.
The VCM operates within a negative feedback closed loop control system.
The VCM consists of an actuator arm pivoted to base casting with a pair of
ball bearings, voice coil and suspension. The VCM has a large stroke and
a relatively coarse resolution and the head can be rotated across the disc.
3
Next generation actuators, called microactuators are used to extend band
width capability of a single stage actuator. Increasing bandwidth capability
enables increased track pitch, and thus increased data storage capacity.
PiezoElectric Transducers (PZT) are used to control the micro actuator
motion similarly to the voice coil motor in the classical single stage design,
except with limited stroke capability. The PZT converts electricity into
mechanical motion. Large displacements or forces are generally not avail
able with Piezo film. However, the PZT can be used to excite mechanical
structures over a wide frequency range. The output of mechanical mo
tion and dynamic range is proportional to the volume of the film stressed.
Film thickness may be chosen to optimize the electrical signal format or
to respond to mechanical strength requirements. The standard single stage
Voice Coil Motor control loop bandwidth ranges from 500 to 1500 Hz. How
ever, when the Piezoelectric micro actuator is mounted on the tip of the
arm, suspension or head allows for much higher bandwidth capability, in
the range of 2 to 5 kHz.
Various micro actuator designs are being developed to overcome the chal
lenges related to the coupling of the vibrations caused by either the Piezo
excitation or the available dynamic range. High frequency suspension modes
are lightly damped and can be easily excited by the micro actuation control
signal or other external disturbances, such as airflow. Controllable motion
with the minimum resonance excitation is the key goal for any micro ac
tuator design. The resulting control system involves single position error
signal output and multiple control signal input. The interaction between
4
twocontrol inputs and the mechanical structural motion often results in
some performance degradation.
This section summaries the types of micro actuators and their design trade
offs in terms of designing a servo loop.
1.1.2.1 Arm Based Micro Actuator
One of the design tradeoffs common in most micro actuator designs is the
location of the Piezo device for the implementation of rotary motion result
ing from micro actuation. To achieve the stroke that can meet the controller
dynamic range without saturation, a Piezo device with small displacement
requires leverage or mechanical advantage. An arm based actuator maxi
mizes this benefit by placing the Piezo device in the arm, thus significantly
amplifying the rotary motion. However, enormous challenges then arise due
to the lower arm stiffness and the coupling of higher frequency arm dynam
ics with suspension; the result is then a overall degraded performance for
the dual stage controller.
1.1.2.2 Suspension Based Micro Actuator
A suspension based micro actuator minimizes some of the disadvantages
of the arm based micro actuator in terms of structural stiffness. However,
coupling of the arm and suspension resonances are possible, unless the de
sign is optimized to decouple these resonance frequencies. Piezo devices are
mounted on the. suspensions; usually two Piezo electric motors compensate
for the loss of the mechanical advantage provided by the arm based micro
5
actuator. The key concerns in this design are nonoperational shock, fa
tigue due to ramp load and unload and variation of Piezo displacements
with respect to the temperature. In addition, high frequency suspension
resonance is lightly damped and acts as a spring, resulting in significant
increase of high frequency nonrepeatable run out.
1.1.2.3 Slider Based Micro Actuator
Slider Based Micro Actuators are also called as colocated micro actuators.
A low frequency mode of slider gimbal assembly provides significant advan
tage in terms of decoupling micro actuator dynamics and voice coil arm
modes. Availability of dynamics range is minimum in this type of design.
Improved multilayer Piezo technology offers advantage in maximizing the
displacement with minimum voltage.
1.1.2.4 Micro Machined Actuator
This is a relatively new and under development technology involving in
tegrated head, actuator and slider concepts. This micro actuator is batch
fabricated on a sacrificial layer of wafer together with an electrostatic mi
cro actuator. This dual stage controller technology is not influenced by
the mechanical structural resonance of suspension or arm, since the macro
actuator is part of the slider that includes air bearing. Apart from the man
ufacturing process itself, the biggest technological limitation is the available
dynamic range, however fine head positioning may be possible.
6
1.2
Key Components of a Digital Servo
Disc drive is one of the most complex practical applications for various
types of control systems. Disc drive controllers include tracking, seeking and
spin controls are required to meet demanding performance characteristics.
Various complex theories such as robust and optimal control, adaptive as
well as classical, are used to design and implement various controllers in
a disc drive. Figure (1.2) shows the solved finite element model, used to
obtain structural dynamics and plant. Model obtained from such analysis
is used to simulate the performance of the disc drive digital control system
performance.
1.2.1 Seek
A moving actuator with a read and write head, from one data track to
another data track is called a seek. Reaching a destination track as fast
as possible is the typical seek strategy. A seek strategy involves various
design issues, such as switching points from acceleration to deceleration
and the implementation of open loop vs closed loop controls. Longer seeks
typically use a velocity mode control, to minimize the memory requirements.
Short seeks are called model reference or position mode seeks. A servo
sector contains information to uniquely identify each track; is called gray
code. Seek uses the gray code information to find the destination track and
the distance to go. Seek consists of three phases: the acceleration phase
that initializes the actuator motion and picks up the speed; the cost phase
7
required to limit the maximum velocity and generally existing in longer
seeks; the deceleration phase, when the actuator slow down and arrives on
the desired track. The arrival part can have the most profound impact on
the performance of seek and data transfer rates of the disc drive. Seeking
is one of the most expensive operations performed by a disc drive in terms
of performance and cost. It is almost impossible to stop instantaneously
a high velocity moving actuator on the arrival data track; overshoots and
oscillations are common problems resulting in performance degradation.
This thesis includes fundamentals of seek and feed forward seek controllers,
and investigates the solutions for some of the seek issues.
1.2.2 Settle
A transition between seek and tracking is called settle. A settle control
allows the initialization of a tracking controller. The key considerations
in the settle control design are minimizing either the number of samples
or the settle time and ensuring a smooth transition from the settle to the
tracking controller. In its simplest form, the settle controller can be same
as the tracking controller, except the integrator action is turned off during
high velocity samples, to minimize overshoots caused by strong integra
tor gains. Controller switching and initialization issues are avoided with
blending schemes or nonlinear control actions. The tracking controller in
tegrator action is turned off by adding zero at some frequency, by moving
the zero to a high frequency where increasing integrator gain can minimize
the settle time. Settle can also have profound effect on the performance of
8
the drive data transfer, especially during external vibrations. In addition,
the tendency of going off track after switching to a tracking mode is key
consideration for designing an optimal settle controller. A micro actuator
can minimize settle time by a significant percentage.
1.2.3 Tracking
Accurate positioning of the read and write head over a data track on the
disc is the most important for reading or writing the data. Track densities
of disc drives have been continuously increasing over many years. Higher
track density requires more accurate and higher resolution positioning. Sig
nificant research efforts have been involved to extend capacity and track
densities of disc drives. Tracking control has to deal with lower servo sam
ple rates, structural modes, external shocks, vibrations, written in position
errors, pivot bearing nonlinearity, gain variation from magnetic transducer,
cross track non linearity and the effects from various noise sources resulting
in erroneous position signal. A conventional disc drive tracking controller
includes ProportionalIntegralDerivative (PID) or State Feed Back (SFB)
controller. However, modern disc drives include multivariable state space
or adaptive controllers. A servo actuator control includes a single stage
single input single output system or a dual stage multiple input multiple
output system. Tracking control also includes external disturbance cancel
lation using accelerometers. A key objective of a tracking control is the
minimization of Track MissRegistration (TMR). To implement a discrete
time controller with fixed point digital processor and limited processing
9
power, number of additional challenges needs to be overcome. If a sensor
based disturbance information exists, then the system performance can be
improved by matching the tracking feed forward filter dynamics and the
system dynamics. By selecting the filter dynamics to match the tracking
system, both systems may have the same response. In this thesis, efforts are
made to derive a simple feed forward algorithm to minimize external dis
turbance. Practical implementation issues and probable solutions are also
discussed. The first part of this thesis deals with system modelling and var
ious transfer function identifications. A compensation scheme that uses a
fixed parameter is developed. Low cost disc drive requirements imposes al
gorithmic simplicity in the actual implementation. A tracking controller for
a disc drive requires minimization of position error, to protect data integrity
and achieve the targeted track pitch. Designing a servo controller of a disc
drive with ever increasing track pitch density is a challenging problem. Min
imization of track missregistration is achieved by increasing control loop
bandwidth. Structural resonance of head stack assemblies is one of the
major limiting factors towards servotracking performance in higher head
positioning. The first significant resonance usually called as system mode
results 180 degrees of phase loss. Loss of structural stiffness at frequen
cies higher than system mode limits the control loop bandwidth. Lightly
damped other modes can be excited easily by the airflow, are mostly sta
bility and Track MissRegistration concerns. The control system command
current and other external as well as internal resonance provides excita
tion sources for any structural resonance that can affect the performance
10
and robustness. Active control of these vibrations requires high sampling
frequency and servo loop bandwidth. However sampling frequency of the
position error signal is limited by the number of the servo sectors written on
the disc. Increasing the number of servo sectors minimizes the space avail
able to store data on the magnetic disc. Voice coil motor assembly provides
low speed rotary motion over a large stroke length, however with limited
effectiveness, regarding rejection of high frequency resonance. Continuous
efforts to move the structural resonance at higher frequencies is represents
typical strategy for enabling higher bandwidth and thus higher track pitch
density. To enhance the capacity of a disc drive requires the ability of fine
positioning in highspeed motion. A micro actuator provides this highspeed
motion with limited stroke. Combination of low speed voice coil motion and
highspeed micro actuator motion enables high bandwidth capability. This
thesis explores single stage and dual stage state space control design with
focus on advanced multivariable and robust control theory.
1.2.4 Spin
A disc drive spindle motor spin speed is maintained constantly by a closed
loop servo controller. The control action is based on measured Back Electro
Motive Force (BEMF). The spindle motor controllers has relatively low
bandwidth loops, some where from 10 to 50 Hz. A motor driver incorporates
advanced features to optimize spin control, such as Smooth Drive Digital
Architecture (SDDA). A spindle motor driver provides the motor control
using nchannel MetalOxideSemiFieldEffectTransistor.
11
Figure 1.1: Finite element model of disk drive
12
{
Figure 1.2: Finite element solution of disk drive
13
2. Model Development
There are many ways to design a feedback controller. With the goal of
achieving maximum disturbance rejection and considering the shortcom
ings of classical and LinearQuadraticGaussian (LQG) control theory, this
analysis synthesizes controller using %<*, optimization and //synthesis and
ensures robust control as well as maximum disturbance rejection. The inten
sion here is not to provide a detailed mathematical solution, since efficient
commercial software is available for the pertinent computational objective.
We rather seek to provide a generic problem formulation which might be
useful to solve disc drive control design problems. The modern approach to
characterizing closedloop performance objectives is to measure the size of
a closed loop transfer function matrix using various matrix norms. A ma
trix norm provides a measure of the magnitudes of output signals triggered
by certain classes of input signals. Optimizing these types of performance
objectives over the set of stabilizing controllers is the main thrust of recent
optimal control theory, such as L\, H2, and //synthesis. It is thus
important to develop clear understanding of how many types of control ob
jectives can be posed as a minimization of closed loop transfer functions or
norms. In this thesis we consider a general model for formulating a control
14
problem introduced by Doyle [1] described in Chapter (4).This formula
tion makes use of general control configuration where P is the generalized
plant and K is the generalized controller. The overall control objective is to
minimize some norm of w to z, for example the 'Hoa norm. The controller
design problem translates then, to finding controller K which is based on
the information in v, generates a control signal u, which encounters the
influence of w on z, thereby minimizing the closed loop norm from w to 2.
High volume manufacturing also introduces another challenge for design
ing a robust control solution. Part to part variation as well as variation
in actuator dynamics could affect stability and performance, if the control
loop is not designed carefully. H00 and /isynthesis allows to specify accept
able perturbation. However, understanding the plant dynamics variation
also represents fundamental aspect in a control design problem formulation.
This chapter not only develops a methodical approach towards development
of a plat model that can be used to design a seek, settle or tracking control,
but also presents a variation analysis methodology based on finite element
analysis. A plant model with unstructured uncertainty bounds is shown in
Figure (2.1) for a single stage disc drive.
2.1 Obtaining the Generalized Plant
In model based control it is assumed that the plant knowledge is sufficient
to design a controller. However, effectiveness of the controller synthesis is
limited by the plant knowledge as well as by the ability to define reasonable
control objectives. A classical controller approach such as a Proportional
15
IntegralDerivative Controller has conserved the popularity among the con
trol community due to its online tuning approach [2]. This section provides
the methodology for a generalized parameterized plant that can be used for
control design and analysis.
2.1.1 Single Stage Analytical Plant Model
Open loop transfer function of single stage control
i9siso ~ K'
VCM Gvcm
(2.1)
where Gvcm{s) is a Voice Coil Motor transfer function and KVcm is the
controller with closed loop and sensitivity transfer functions as shown below
G,
clPs
i + G and G" ~ 1 + a,
(2.2)
rig 1 I lg
Gvcm{s) is expected to accurately characterize the nominal performance of
actuator. The nominal model should represent the average expected system
behavior seen over entire class or population of actuators being modelled:
G\
K
n _2
DC
TVCM s2 + 2Qion + ul s2 + 2CP,iUnp>i + ^lp .
where KDC is DC gain of the actuator, a; is natural frequency of mode i =
1,2,3, ,n and ( is damping, sub 2 and p to differentiate poles and zeros.
Since continuous time Voice Coil Motor is sampled at the sample rate, the
measured frequency response includes transport as well as computational
delay. If the total delay assumed to be T seconds, the Laplace transform of
time delay is e~sT. This exponential transfer function can be approximated
S +2Cz,jWnt,i + Vnx,i
(2.3)
16
in the form of a rational transfer function using Pade approximation [2].
Â£{xttd }
9 _L_ V'n (td3)'
Z 2Lq=0 j\
9 _L_ X^n (tdS)'
L "T Z^i=0 i\
(2.4)
where td is the delay, s is the frequency parameter and Â£{ } argument
represents Laplace transform. If Ts is sample time then,
td ~ tfransport T tcompUtational +
(2.6)
Critical aspect of digital control design is to understand sample hold effects,
phase loss due to hold effects has significant impact on the controller phase
margin. For a continuous time signal f(t) = e~at where t > 0 corresponds
to a pole at s = a, which has Laplace transform,
' <26>
and the z transform of f(kT) or F(z) = C{e~akT} is equivalent to
corresponds to a pole at z = e~aT. This means that a pole at s a in
splane corresponds to a pole a.t z e~aT in discrete domain. The bilinear
transform is a common way to convert continuous time transfer function
to discrete time. The bilinear transform is performed quite literally by
replacing every occurrence of s in a H (s) by
s =
2 (1 z~l)
Tsample (1 T %
so H(z) = ff(z)
ii
(2.8)
^sample (1+z
The bilinear transform is also called as Tustins approximation [2]. The
zero order hold transfer function is defined as
ZOH(s) =
1
nplejM
nplej ^
(2.9)
17
ZOH phase loss at cross over or control loop band width frequency can be
approximated by
ZOHphaseL0ss(Deg)
180
BWFrequency{Hz) ~pi~^
90
= BWFrequency(Hz)^ (2.10)
where Fs is the sample frequency can be calculated by multiplying disc
drive RPM and number of servo wedges or sectors.
2.1.2 Dual Stage Analytical Plant Model
There are many ways to model the dynamics of a dual stage actuator that
involves some way of decoupling the control and sequential designs of mul
tiple single stage components. Depending upon the interactions between
micro actuator and voice coil motor, the two control loops can be treated
separately or combined. Decoupling minimizes the performance loss. How
ever, stability can be a key concern compared to coupled design. The plant
transfer function for dual stage servo controller is modelled as shown in
Figure (2.2)
G(s) = [GVcm(s) GMicro(s)] (2.11)
where Gvcm(s) is VCM and GMicro is the micro actuator transfer matrix,
and where
K(s)
Kvcm(s) FVCm(1 T FMiCr0GMicro) (s)
or
FMicro(s) FMicro(s)
(2.12)
18
where Kvcm(s) is VCM and KMicro is the micro actuator controller trans
fer matrix. Based on equation (2.12) the open loop transfer function (i.e.
without pes feedback loop being closed from r to pes) is:
Gig KvCM GvcM + KMicro Guicro (213)
The closed loop and sensitivity transfer functions are given as follows:
Gap 1 Gllt and Gsen = 1 (2.14)
i + trIg i + &lg
The open loop transfer function for the decoupled controller is
Gig = KVCM GvcM + KMicro GMicro
+ [KMicro SpztKvCM GvCm] (2.15)
For most of the Piezo actuated suspensions, the estimated Relative Position
Error Signal can be obtained by multiplying a DC gain term with the micro
actuator control input. Assuming that the Relative Position Error Signal
can be perfectly estimated, we then have,
Glgideai KvCM GvcM + KMicro GMicro
+ [KMicro GMicroKyCM GvCm\ (2.16)
By adding one to both sides, we obtain
1 + Gigidea[ (1 + KvCM GvcM ) (1 + KMicro GMicro) (217)
Thus, the ideal close loop sensitivity transfer function is,
1 1 1
1 + Gigideal (1 + KVCM GvCM ) (1 + KMicro GMicro)
(2.18)
19
the ideal sensitivity for the dual stage control loop is the product of VCM
and the Micro Actuator sensitivity transfer functions:
Sideal SyCM SMicro
(2.19)
The dual stage control system is stable if both VCM and Micro actuator
loops are stable and the control loop design problem can be decoupled
into two independent controllers. The VCM loop can be designed by using
conventional single stage controller; the micro actuator controller can be
designed to increase bandwidth and error rejection. For most of the micro
actuators, the low frequency response can be approximated by its DC gain.
Thus, gmicro = G Micro + AG Micro where AG Micro is small compared to
GMicro However, in the high frequency range the resonance modes of the
micro actuator dominate the frequency response of GMicro So, to ensure
the robust performance, we have:
KMicroAGMicroKvCM << 1 + KvCmGvCM + KMicroGMicro+
KMicroGMicroKvCMGvCM \ (2.20)
Therefore, the approximate decoupling can be achieved. Formulation of an
accurate mathematical model of a dual stage control represents a first step
to the synthesis and to implementation of a controller via Hoo or /xsynthesis.
It is common to use a simpler multiple input single output model in order to
simplify the controller design problem. However, it is important to develop
a relatively accurate model, to represent enough flexible dynamics of voice
20
coil motor and micro actuator:
G(s)
Ap Bp
cv 0
with a VCM model
Gvcm(s)
a micro actuator model
Avcm 0 Bvcm 0
0 AMicro Bvm BMicro
Gvcm GMicro 0 0
(2.21)
Avcm Bvcm 0
Gvcm 0 0
(2.22)
G
Micro
() =
AMicro Bmv BMicro
GMicro 0 0
(2.23)
The Bmv represents the excitation of micro actuator dynamics due to the
voice coil motor. In simplified terms,
D D KtvCM KtMicro
Â£>mv &Micro T T
JVCM J Micro
(2.24)
Where KtMicro and KtycM are torque constants of micro actuator and
voice coil motor and J Micro and Jvcm are the inertia terms. Therefore for
the Voice Coil Motor; we have:
Avcm 0 AycMi2 j BVcm 0
0 0 (Jvcm T lmicro) KtvCM / JvCM
a
VCM
Gvcmi 0
(2.25)
21
for the micro actuator we have:
Micro
0
'Micro
,Cvcm= Cmicro. 0 (2.26)
where lvcm and lmicro are effective arm lengths for VCM and micro actuator,
and kmicro and bmicro are stiffness and damping factor for micro actuator.
The contribution of Bmv becomes negligible in case
{KtyCM JMicro)/i/Kt Micro JvCm) ^^ (^inicvo The detail discussion of
developing a plant model will come later in section (2.2).
2.1.3 Seek Parametric Plant Model
Following the discussion, we have:
Where Kt is the Torque Constant measured in inozf /A, Larm is the length
of the Arm or the Pivot to Head distance measured on inch, TPI is tracks
per inch, Tsampie is sample rate obtained by 1 /RPM/NumberofServoSector
and J is the actuator inertia measured in inch oz sec2. So the units of
where i is the current in Amp, n is number of coil turns, L is length of
active part of the coil or the radial section length in inch, B is magnetic
J
(2.27)
Kact becomes tracks/sample2/Amp. Also following definitions, we have:
Torque = Kt i and Kt = n L B r
(2.28)
22
field in gauss and r is the average radius of active part of the coil in inch:
DACgain = ^ (2.29)
and
ts ___ _____max_______
pa ~ 2(DACtn1) 1
(2.30)
where Kpa is power amplifier gain in terms of Amp/Bit, Imax is maximum
coil current and DACuts is the DAC size or resolution in bits. Note,
Sly cm ;x77: and DIvcm = Sly cm Sly cm (231)
S + 2 7T 10 v
and
Gvcm = DIvcm Kact Kpa (7?,Ka (2.32)
\ J sample /
2.1.4 Transconductance Loop Design
The typical disc drive VCM driver uses Nchannel MOSFET to implement
a transconductance class amplifier. This amplifier consists of an error
amplifier, a drive amplifier and current sense feedback amplifier. A H
bridge driver is implemented in the VCM driver. The current command
voltage is the output of an error amplifier as the summing point for the
transconductance control loop. The summing point includes the outputs
of the VCM current command, called DAC, and the feedback from the sense
amplifier:
7y _____ Icoil
Rfb
RinRsenseKsense
(2.33)
23
2.2
Modelling High Frequency Dynamics
A three degree of freedom model is given by
o o 1 x\
o o Â£2 +
1 o o 1 X3
k k 0 xi F\
k 2k k x2 = F2 (234)
0 k k xs F3
the eigenvalues axe w\ = 0, w2 = y^ and in the physical
coordinate system where equations are coupled and have to be solved:
mxi + kx i kx 2 = Fi
mx2 kxi + 2kx2 kx 3 = F2 (2.35)
mx3 kx2 + kx3 F2
example of principle coordinate system where homogeneous equations are
uncoupled and can be solved independently:
xpi = 0
xP2 + ^xp2 = 0 (2.36)
+ f = 0
Eigenvalues and eigenvectors can be normalized with respect to mass, above
equations can be rewritten in the motion coordinate system. The forces in
the principle coordinates Fpi,Fp2 and Fp3 obtained by premultiplying the
force vector in physical coordinates by
Fp = xiF
(2.37)
24
The three equation of motion in principle coordinate becomes
Â£pi Fpi
%p2 + 2(2l^2ip2 + ^2Xp2 Fp2 (2.38)
^p3 + 2C3^3^p3 + w3^p3 = Fp 3
where u>i,u>2 and u3 are the three eigenvalues with units in radians/sec. (j,
C2 and C3 represents percentage of critical damping. Next step is to convert
these equations in state space format by solving the equation for highest
derivative
xpi Fpi
xp2 = Fp2 2(2U2Xp2 u\xP2 (2.39)
xP3 = Fp, 2Ct,u),xv, 0J3xp3
so the states
Xi = Xpi Displacement of mode 1, not the mass 1
X2 = xpi Velocity of mode 1
x, = xV2 Displacement o f mode 2
(2.40)
X4 = xP2 Velocity of mode 1
X5 = xp3 Displacement of mode 3
x6 = ip. Velocity of mode 3
rewriting the equation of motion in the state space [3] form:
25
Xi x2
X2 Fp 1
X3 = xi
X4 = Fp2 CJ2X3 2(2CJ2X4
Xe = Fp2 wx3 2C2W2X4
(2.41)
The state update equation x = Ax + Bu in matrix form
x'l 0 1 0 0
X2 0 0 0 0
X3 0 0 0 1
Â£4 0 0 2^2^2
Â£5 0 0 0 0
x6 0 0 0 0
0 0 Xl 0
0 0 x2 Fpi
0 0 x3 + 0
0 0 Â£4 FP2
0 1 Â£5 0
, ,2 2C3u>3 Â£6 Fp3
(2.42)
for physical structures direct transmission gain for zeros is zero so the feed
forward D term can be ignored in the output equation y Cx Du.
Since output of disc drive actuator is displacement, the output equation yp
becomes
Vp
Cx =
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
 "
Â£1 Â£1
X2 X2
X3 X3
Â£4 Â£4
Â£5 Â£5
Â£6 Â£6
(2.43)
26
With the desired outputs in position and velocity in principle coordinates,
next step is to back transform them to physical coordinates by
EnUp
Â£Till 0 nl2 0 ^nl3 0 Upi
0 ^nll 0 ^nl2 0 nl3 Vp2
Â£ji21 0 Xn22 0 Â£n23 0 yP 3
0 71.21 0 Xn22 0 %n23 Upi
%n31 0 %n32 0 ^'77,33 0 Vp5
0 Zn31 0 Xn32 0 Â£71.33 Vp6
(2.44)
Figure (2.7) shows frequency response of such model that can be useful to
analyze control system including Monte Carlo and whatif analysis.
2.3 Estimator Model
Direct velocity measurement is not available for velocity mode seek con
troller. The feed back is generated with the velocity estimator [5] based
on position and the acceleration command. Position is measured at servo
sample, computational delay or time required to compute the position at
servo sector as well as the time required to generate acceleration command
or control effort results error in the expected and actual performance. The
relationship between position, velocity and acceleration are given by:
v(t) = v0 + a(t) t (2.45)
x(t) = x0 + vt t + a(t) t
2
(2.46)
27
Position is measured at time n, however acceleration command is not gen
erated until n + cd. Following are the estimated velocity and position equa
tions:
Vp(n + 1) = v(n) + a(n 1 )cd + a(n)(l cd) (2.47)
Xp(n + 1) = x(n) + v(n) cd + a(n 1) cd
Z
+
1
v(n + cd)( 1 cd) + ^a(n)(l cd,)2
(2.48)
where Vp and Xp are predicted position and velocity, v(n) is the velocity
estimates that represents actuator velocity. If x(n) is measured position
and cd is the code delay then
v(n + cd) v(n) t a(n 1 )cd,
(2.49)
and position Xp becomes
Xp(n + 1) = x(n) + v(n) + a(n 1) led + a(n)(l cd)2 J .(2.50)
Since position x{n) is measured at time instance n and can be used to correct
for any error in predicted position Xp(n). Difference between measured
position and predicted position is defined as Residual RS:
RS(n) = x(n) Xp(n), (2.51)
therefore the corrected velocity becomes
Vc = Vv L RS(n)
(2.52)
28
Gain L can used to correct any error or transients. Vc(n) can be the actuator
velocity at time n. The reason to derive Kiurnp is to allow the input to the
estimator block to. be the output of the controller icmd in (bits) so the
predicted position and velocity equations become:
CV 1 = (1 cd)Kiump (2.53)
CV 2 cd/ Plum.]) (2.54)
CPI = 0.5 (1 cdfKlump (2.55)
CP2 = (1 o.5 cd2)Klump (2.56)
So with these constants estimator equations can be rewritten
Vp(n + 1) = Vc{n) + CV2 icmd{n 1) + CV1 w(n) (2.57)
Xp{n + 1) = X{n) + Vc(n) + CPI icmd(n 1) + CP2 icmd{n) (2.58)
In disk drive tracking control scheme the loop output is position error signal
and seek control scheme is distance to go to reach the desired track. Accel
eration command drives actuator closer to reference track that minimizes
position error.
29
VCM with additive uncertainty response
VCM additive uncertainty wt
x 10
VCM additive uncertainty bounds
Figure 2.1: Single stage plant response
30
Phase [Deg] Magnitude [dB]
Nominal VCM Plant with Nominal Micro Actuator Plant
Figure 2.2: Dual stage plant response
>Q
K_vcm j
VCM Controller
K_micro 
G_vcm
Micro Act Controller
Figure 2.3: Block diagram with dual stage with parallel structure
31
VPES
Figure 2.4: Block diagram with dual stage with decoupled structure
Figure 2.5: Decoupled plant
Figure 2.6: Simplified plant
32
Magnitude (dB)
Figure 2.7: State space model based on FEM
33
3. Velocity Mode Seek Control
Seek controllers [4, 5] are evolved in two fundamental types have their own .
advantages, limitations and uses. First approach is called position reference
seek also called as model reference seek. Seek from one track to another
track is performed by generating a feed forward acceleration profile, posi
tion reference profile is generated by using plant model and error signal is
generated from deviation of desired position trajectory. Error minimization
is achieved by state space, PID or state feedback control and then added
to feed forward signal to form the composite control input. Position mode
seek requires predefined acceleration profile to seek desired track. This pro
file is generated form trapezoidal acceleration and deceleration profile, also
sometimes filtered to minimize jerk. Length and shape of the profile is
determined by seek length and generated acceleration profile is applied by
open loop to the plant. The position mode seeks are optimized by chang
ing shape and amplitude of acceleration profile to minimize the seek time.
Once optimized, position trajectory is obtained by applying ideal acceler
ation profile directly to the plant. One of the limitation of position mode
seeks is memory requirement to store different acceleration and amplitude
profile for each seek length requiring multiple tables in the firmware to
34
store each acceleration profile. Single profile can be used for a group of seek
lengths via interpolation but as length of the seek increases, number of table
entries goes up so position mode seeks are used only for short seek lengths.
Also if the tracking controller is used to minimize error, then dynamic range
of tracking controller can minimize the max seek length mainly due to the
increased error for longer seeks. Figure (3.1) shows block diagram for a typ
ical position mode seek control. Second approach is called velocity mode
Figure 3.1: Position mode seek
seeks. Velocity mode seek as shown in Figure (3.2) is performed by generat
ing a desired velocity profile or reference trajectory and then calculating the
respective acceleration command by using Newtons Physics: Velocity mea
surement is directly unavailable with disc drive plant, so velocity estimator
computes the estimated velocity. Difference between estimated velocity and
35
velocity reference used to generate a velocity error signal and gained error
signal with a feed forward term are combined to generate control input to
Position error signal is the only measurable output available for disc drive
actuator. Since, acceleration is second derivative of position error and plant
control input is acceleration command, defining desired acceleration profile
could provide better control over position during a seek, so defining the
acceleration profile is first step in developing velocity mode seek control
algorithm. Output of seek controller is current proportional to accelera
tion command thus goal is to obtain acceleration profile which will provide
corresponding velocity trajectory. If the velocity of actuator can be esti
mated based on the measured position then the generated corresponding
velocity can be subtracted from predicted velocity to generate velocity error
signal. Constant acceleration and deceleration is used to develop velocity
mode seek profile. Based on Newtonian physics, the relationship between
acceleration, velocity and position:
the plant. Generally velocity mode seeks are used for longer seeks while
position mode seeks are used for shorter ones.
3.1 Deriving Seek Control Algorithm
(3.1)
for a zero initial condition
(3.2)
36
or
t=i
a,o
Once velocity is known as a function of acceleration command and actuator
position, then reference velocity can be generated for velocity control loop
and for defined acceleration command, initial velocity error signal would be
large enough to accelerate the actuator until velocity intercept the square
root deceleration profile. From this point velocity reference would be deter
mined solely by measuring the position. The input to the plant would be a
constant deceleration command plus a gained velocity error signal. During
initial part of the seek control law is dominated by velocity control loop
as velocity error is significantly larger when the actuator begins to move
from zero initial condition. As the actuator velocity increases, velocity er
ror becomes small. For a small velocity error, feed forward term becomes
dominant part in control law and helps to minimize transients. The velocity
trajectory is steep at the beginning of seek as well as when approaches the
target track position. Phase plane as shown in Figure (3.3) and Figure (3.4)
is x y plot and one axis is derivative of other axis. It is educational to
look at two phase planes for seek analysis, velocity vs. position and accel
eration vs. velocity. Phase plane analysis is a common way to understand
the rate of change of one variable with respect to other. Since velocity is
derivative of position phase plane shows how fast the actuator moves for
a given position. Also maximum acceleration and velocity of the actuator
is controller by saturation commands. The saturation plays bigger role for
longer seeks by adding a constant velocity or cost phase. Velocity reference
37
based on measured position and acceleration command by substituting t in
equation (3.2)
(3.4)
therefore v in terms of square root law becomes
v(t)2 = 2 a;(t)a0, and v(t) = \/2 x(t)a0
(3.5)
3.1.1 Exponential Arrival and Switch Point
Actuator inertia and torque constant imposes limitations upon how fast
actuator can move for a applied acceleration command. Due to a large ac
celeration command requiring rapid change in the velocity profile, actuator
lags behind desired velocity and velocity error keeps growing to the extent
increasingly becomes difficult to stay on the profile. This causes a large
velocity error as seek approaches the desired track. Also when constant
deceleration current applied during seeks is discontinuous at the reference
track resulting a large transient in current command at the end of seek. This
makes the practical implementation of this profile nearly impossible. So it
becomes key to find less stringent arrival profile and deceleration require
ments that can be used when seek approaches to reference track. Ideally
the desired profile would allow position, velocity and acceleration close to
zero at the end of seek. Since all three signals are related by a derivative,
so the ideal profile can be realized using an exponential or linear velocity
reference. If the exponential arrival defined by a time constant t\ then
38
relationship between acceleration, velocity and position can be defined as
a(t) = a0e v(t) = a0re x(t) = o0r2e
2
(3.6)
and velocity reference and deceleration command as function of position x
(3.7)
(*) = Â£&, and a{t) = vJA
T
SO,
and,
rr ____ rr , SVexp &V,
*exp sqrt 0>^d
sqrt
st
st
$Vexp __ fi^exp QYld sqrt
5x 5t an St
(3.8)
(3.9)
St 5x St St 5x St
For exponential arrival the seek controller must generate velocity reference
and feed forward deceleration command that is similar to square root state
during velocity reference and feed forward command. A square root profile
and an exponential arrival is twofold if the discontinuity presences between
the two trajectories at the switch point. The goal is to derive a seamless
transfer between square root and exponential states. Requirement for a
bump less transfer could be the same position, velocity, and acceleration
at the switch point. This means at particular position velocity reference
of square root state is. equal to the velocity reference of the exponential
state and the deceleration command of square root state is equal to the
deceleration command of the exponential state, where Sx/St is velocity for
each profile:
SVexp __ fiVsqrt ^
Sx
Sx
39
To find a switching point where velocity and acceleration are the same,
square root trajectory need modification to include a position offset xos\
Vsqrt = y/2ao [x(t) Xos(t)] (3.11)
Where ao is feed forward acceleration or deceleration command, x is the
distance to go, and xos is fixed offset. Exponential time constant is defined
as r and its inverse defined as Asched'
ASched = > (3.12)
r
therefore the exponential seek state equations are:
kea',p ASched % (111(1 OeXp Asched ^ (3.13)
Defining inverse of r makes the math little bit simpler by avoiding the
division. If Xs is the switch point, then
Vea:p = Asched Xs CLTld Vsqrt (Xs Xos)] (314)
right sides of above equations to Vexp = Vsqrt
Asched = a/2 Oq(Xs Xos), (3.15)
and by taking square of both sides:
(Asched Xsf = 2 a0(Xs  X0S). (3.16)
therefore the inverse and square of time constant r in terms of switch point
and offset are:
= 2ao(*J~H (3.17)
(Wsj
40
where the velocity condition
&Vexp
5x
= A
sched
j &Vsqrt
and
OX
1 2 a0
2 V {X, xos)
(3.18)
The goal is to find xos in terms of Xs, and to derive a closed solution, so
the switch point in terms of acceleration and time constant becomes
a o
(^sc/ied)^
(3.19)
Above equation allows seek profile to follow the square root trajectory to the
switch point Xs and then follow exponential trajectory with seamless tran
sition between velocity and acceleration. Also the switch point is function
of inverse exponential time constant, Asched and the square root decelera
tion constant, o. Seek then can be designed by choosing the deceleration
constant ao for square root state and time constant, r for exponential ar
rival. The advantage of this technique can be seen in the phase plane results
at reference track both the position and velocity are approaching to zero
with reasonable rate. Since deceleration is proportional the coil current,
also goes to zero as seek approaches to reference track thus avoiding large
change in the coil current. Next step is to determine the desired deceleration
constant.
VMff = p (3.20)
lump
Figure (3.5) shows the feed forward term with respect to seek acceleration
profile. The velocity reference is generated and compared to velocity esti
mate. The difference between required velocity and actual velocity is called
velocity error signal. The velocity error is multiplied by the proportional
41
loop gain Ksgrt and added to the feed forward term to generate control
signal:
and, iVReS V^jKiump V MFF whcTG V/jej sj
Since the feed forward term the deceleration command it is subtracted
from the gained velocity error term. With the large initial velocity error
current command will be positive and accelerate the actuator towards the
reference track. As the velocity approaches the desired reference profile
the deceleration feed forward term starts to dominate and the actuator will
decelerate along the desired trajectory. The constant current deceleration
profile will be followed until the distance to go reaches the switch point Xs.
After this point, seek profile control equations switch to exponential profile.
Velocity reference, feed forward and control output signal will be generated
for the exponential state where the velocity reference is given by
VRef = AschedXs and VMFF = (3.22)
Ltlump
resulting change in controller output by replacing loop gain for exponential
state:
icmd = (VRef V)Kexp VMff (3.23)
3.2 Addition of Back EMF
Due to the large velocity error acceleration current ramps up velocity of
actuator and effects of BEMF increases so available current to ramp up the
42
actuator decreases. If the cost states exist, current being driven is very small
or close to zero so velocity will remain at the max velocity limit. Once the
actuator starts to approach the deceleration profile velocity error becomes
small compared to feed forward deceleration command. The current in the
coil starts to follow the feed forward command. This can be seen by the con
stant current applied during the deceleration section of seek. One necessary
addition to the profile generator is required to predict accurate model of
disk drive seek controller is to account the additional voltage available dur
ing deceleration section of seek. During the acceleration section the BEMF
voltage is limiting the available current in the coil by limiting the available
voltage across the coil. Voltage induced during the deceleration section due
to BEMF subtracts the supply voltage and during the acceleration adds to
the supply voltage. So the seek control algorithm can be designed to take
advantage of this phenomenon. BEMF voltage is directly proportional to
the velocity of the actuator and exerts additional current that is available to
decelerate the actuator.If BEMF current is known then feed forward term
of controller can be modified to account change in acceleration or deceler
ation command. So first step is to derive new feed forward command that
includes the change in available current. As BEMF is a function of velocity,
the additional voltage due to BEMF can be defined in terms of Kt. It is
interesting to note that torque constant Kt is the same as the BEMF in SI
units and used to convert current in amps to a torque in Newtonmeters.
By converting Kt in radians/sec to a voltage2 to make the units consistent
43
with the BEMF constant Kemf
K F
Vemf = Kemf uj where Kemf =  (3.24)
L 1 J'arm
Additional voltage based on additional current so the following constant
Kdbemf defined as
= wt (326)
Now it is possible to generate a new feed forward equation for seek square
root profile that takes advantage of additional current due to BEMF.
VMpf = p + V K^ml (3.26)
With new feed forward term includes constant current deceleration com
mand derived earlier and also deceleration current due to the BEMF, ve
locity reference can be modified to include both terms. Square root ve
locity reference command encompasses constant current deceleration based
on position input and generates velocity reference that follows deceleration
command. If the seek follows feed forward deceleration command exactly
then the ideal velocity and position trajectories can be derived analytically.
This can be is done by starting at zero initial conditions and using desired
acceleration profile to build up position velocity profile backwards until the
maximum velocity is reached. This method gives ideal velocity reference
command for any given position. The velocity reference signal generated
based on a constant current deceleration determined by square root of twice
deceleration constant times an offset position to go and feed forward term
increased by a term proportional to velocity. Modified velocity reference
44
equation given by
VRef = y/2 (a0 + KdV)(Xxos)
(3.27)
Where Vprofue is the ideal velocity trajectory computed at a given position
Xprofiie Since Vprofiie is based on modified feed forward term and differs
from velocity reference profile Vsqrt generated from the original constant cur
rent feed forward deceleration command a new term V^f can be defined
as
_____Vprofiie ______
T r Vprofne
VRef err ~  ~
(3.28)
Vsqrt a/2 Qq (Xprofile ^os)
Error between modified and original velocity profile is known at any given
position. Desired profile Vpr0fue approximated by modified velocity refer
ence profile Velref and if matches ideal profile T^ro/iie ratio of Velref to
Vsqrt can be the same as VrefeTr. This provides useful mathematical tool to
obtain Kd for modified velocity reference equation:
VRef \/2 (ap + Kd V) pf Xqs)
Vj
Vsqrt a/2 Q,q (~XprofiXe%os)
By taking square of both sides (3.29)
/T r \2 2 (oo + Kd V) (X xos)
( VRef err ) cy /y _ \
" W'O \/Yprofile os)
and by solving the above equations
(VW.J2 = i + ~ v,
&Q
Rrefe
(3.29)
(3.30)
(3.31)
Next step is to define Kd as a function of velocity V by comparing modified
velocity reference to ideal velocity profile as ratio of errors in the form of
45
a linear slope against a known profile. Therefore above equation can be
redefined in the form of y = mx + b where m, slope is defined SLs Kd/a0.
This allows Kd to be determined by fitting the curve given by (Vreferr)2
with a first order polynomial. The slope, m is determined to be the best fit
will be equal to Kd/a0. Since oo is known then Kd is defined by
Kd = mao (3.32)
Figure (3.7) shows the simulated seek acceleration profile including back
emf effects, position profile, velocity estimate and velocity error and ref
erence and Figure (3.8) overlays velocity estimate, velocity reference and
velocity error to demonstrate high velocity error during initial part if the
seek. Arrival part of the seek has low velocity error resulting feed forward
becoming dominating part of the control signal.
46
Figure 3.2: Velocity mode seek
Position Vs Velocity at the Seek End
Figure 3.3: Position vs velocity
47
Seek length Vs Phase Plane
Figure 3.4: Seek length vs phase plane
Figure 3.5
Feed forward term with respect to acceleration
48
Velocity Acceleration
Figure 3.6: Matlab seek simulation
Acceleration Command xj04 Postlon
Velocity Estimate
Figure 3.7: Velocity mode seek simulation
49
Magnitide
x ) q Seek length Vs Velocity Error
Figure 3.8: Velocity error high at start and low at end
50
4. Track Follow Control
Key design goal of a single stage or dual stage tracking controller for disc
drive actuator is to achieve maximum disturbance rejection. This analysis
synthesizes tracking controller using 'H
and ensures robust control [6] as well as maximum disturbance rejection.
Nominal voice coil motor and micro actuator plants are considered as rigid
body along with, key modal dynamics in order to develop some intuition
and understanding how 'HCX} or //synthesis designs controllers and what it
means in terms of classical control theory. Analysis in this part is used to
develop foundation of more complex control design techniques that includes
part to part variation and designs controller that is capable of achieving
robust performance such as //synthesis. Various types of penalty weights
are discussed and their impacts on the controller and the loop shapes are
also analyzed.
The controllers designed by using Hoo or // theory are used to develop a time
domain simulation that can be used to understand micro actuator stroke
requirements and external vibration performance. How addition of flexible
dynamics can affect the controller design is also discussed in this chapter.
51
4.1 General Control Problem Formulation
The modern approach of characterizing closedloop performance objectives
is to measure the size of certain closed loop transfer function matrix using
various matrix norms. Matrix norm provides a measure of how large out
put signals can get for certain classes input signals. Norm is significantly
inquisitive to a servo engineer who wants to analyze or design control loop
response. Optimizing these types of performance objectives over the set
of stabilizing controllers is the main thrust of recent optimal control the
ory, such as Li, H2, and H^. Hence it is important to develop a clear
understanding of how many types of control objectives can be posed as a
minimization of closed loop transfer functions.
In this section we consider a general model for formulating a control prob
lem introduced by Doyle [1] as shown in Figure (4.1). The formulation
makes use of general control configuration. Where P is the generalized
plant and K is the generalized controller. Overall control objective is to
minimize some norm of w to z, for example 'HO0 norm. The controller de
sign problem is then, find controller K which is based on the information
in v, generates a control signal u, which encounters the influence of w on z,
thereby minimizing the closed loop norm from w to z.
A systems response tendency to grow or decay in time domain characterizers
the system stability [7]. In most general sense, stability refers to charac
teristics of time variation of response. From a practical viewpoint, we may
refer to determine the stability of the system which produces that response.
52
In order to decide whether the response is growing or decaying, a measure
of magnitude or length of a norm must be defined, hence Euclidiean norm
can be introduced as
is useful measure of multivariable systems because it is a positive definite
scaler function, being zero only if all components are zero and diverging if
any single component diverges. The static equilibrium point is evident if
In other words, it is insufficient for the response to be bounded; it must
decay to zero as time passes. This definition of stability is somewhat more
appealing, in that it guarantees that transients will decay. Meeting the
criterion for exponential asymptotic stability assures a given rate of conver
gence, then
lim a;(t) = 0
(42)
Â£(t) < fee Qt x(0), where k, a > 0
(4.3)
then the norm x(t) is guaranteed to lie within a decaying envelope whose
convergence rate is determined by a. Because
(4.4)
integrals of the norm x(t) are bounded
r poo j,
/ P(*)ll*= / \xT(t)x(t)\1/2dt<\\m\\
Jo Jo a
(4.5)
Consequently, if we define
X(t) = D x(t)
(4.7)
such that
Q = DtD > 0
(4.8)
then the integral
poo poo
/ [XT(t)X(t)]df = / [xT(t)DTDx(Â£)]dÂ£ (4.9)
Jo Jo
poo poo
/ [xT(Â£)DTDx(t)]dÂ£ = / [xT(Â£)Qx(Â£)]dÂ£ (410)
Jo Jo
is bounded for stable x(t). The convergence is also true as a bounded
integral of the norm implies stability.
This result has practical significance in the development of optimum and
multivariable control theory.
4.1.1 Signal Norms and Their Properties
Norm of a signals mapped to R piecewise continuous space (oo, oo), may
be zero for t < 0 must have following four properties
IM > 0 positive definiteness
lull = 0 & u(t) = 0 Vt
(4.11)
ait = a M, Va E R
ult> < u + u triangle inequality
54
The 1norm of a signal u(t) is the integral of its absolute value
(4.12)
represents total energy of a signal. Similarly 2norm of a signal u(t)
IMI2
(4.13)
represents power and ooNorm of a signal is the least upper bound of its
upper value
IMIoo := supu(t) (4.14)
t
and for the transfer function G
ll^lloo := sup G(j(u) (4.15)
U>
Then the question of interest would be, for a linear system G with input
u, how big is the output y. H00 norm could be used as a powerful tool to
answer this question.
4.1.2 Obtaining the Generalized Plant
Controller synthesis interconnect [8] is shown in Figure (4.1). The effective
usage of micro actuator is to minimize the high frequency error and the
VCM to track low frequency can complement the micro actuator stroke
and voltage requirements. The plant transfer function for dual stage servo
controller is modelled as
G(s) = [Gvcm{s) G Microns )] (416)
55
where VCM along with zero order hold, computational and structural delay
GyCM Gdelayip) (^bearing () Gsystemic)
KtyCM
STVCM
U
system
JvCM S2 T ^Cbearing^becLTingS + ^^earing S2 + ^Csystem^bearingS T ^'system
(4.17)
and micro actuator
G Micro G Micro (s) Grorsion () Gsway ()
F ^ Mi
Micro
ST]Micro
or
Sway
JMicro S2 + orsion^TorsionS d tOTorsion ^ d" ^Csway^SwayS d" ^%way
(4.18)
Similarly controller can be defined as
Ki^z) \KycM(,z) KMicro(z)\
However the similar controller dynamics can be achieved by
(4.19)
A B
K(z) = CvCM Dvcm
CyCM I^Micro
(4.20)
which has more significance from practical implementation point of view.
Simulation diagram shows the /H00 controller design interconnect. The
closed loop transfer function from w and z is given by linear fractional
transformation
z = Fl{P,K)w
(4.21)
56
where
Fi(P, K) = Pn + P12(I P22K)~lP21 (4.22)
Hoo controller design procedure require solutions to two Riccati equations.
Controller state dimension is equal to the generalized plant. The following
assumptions are made in the problems.
Assumption 1: (A,.B2,C2) is stabilizable and detectable.
Assumption 2: Di2 and D21 have full rank.
A jul B2
Assumption 3:
Assumption 4:
C\ D\2
A jojl Bi
C2 D2i
Assumption 5: Du = 0 and D22 = 0.
has full column rank for all uj.
has full column rank for all u>.
Assumption 6: Di2 =
0
I
and D2i
0 I
In general 1L00 algorithms find a suboptimal controller. That is, for a spec
ified 7 a stabilizing controller is found for which \Fi(P, A) ^ < 7. If an
optimal controller is required the the algorithm can be used iteratively, re
ducing 7 until the minimum value is reached within the given tolerance.
Figure (4.7) shows the synthesis interconnect input and output transfer
functions.
The goal is to convert matrix M as follows
w ~
Mu Mi2
M21 M22
Â£ (j(pi+P2)*{qi+q2)
(4.23)
57
with
Fu(M, Au) = M22 + M12AU(I Mn A2)1M21 (4.24)
provided that (I MnAM) 1 exists.
Linear fractional transforms represents following sets of equation
" 
V = M u
z w
Mu Mi 2 u
1 S rH 1 w
u = Auv
(4.25)
(4.26)
%(X> is a (closed) subspace with the functions that are analytic and bounded
in the open righthalf plane. The Hoo norm is defined as
II C?(s) oo := sup a[G(s)] := sup o[G(jw)] (4.27)
Re(s)> 0 ueR
In this Uoo design problem for disc drive [9] where we need to bound a(S)
for performance,
as noise and a(KS) to penalize large inputs. These requirements can be
combined in to a stacked Hoo problem. K is a stabilizing controller. In
other words z = Nw and the objective is to minimize Hoo norm from w to
In general Hoo algorithms find a suboptimal controller. That is, for a spec
ified 7 a stabilizing controller is found for which \\Fi{P, i^)oo < 7 If an
optimal controller is required the the algorithm can be used iteratively,
reducing 7 until the minimum value is reached within the given tolerance.
58
Weight selection and definition is critical to obtain the optimal solution
and this requires some understanding of tradeoffs and constraints of the
plant. The required sensitivity function is achieved within limits of actuator
control action by maintaining satisfactory gain and phase margins and can
be used as design knob.
4.1.3 Plant Uncertainty
With significant efforts even from experts, no dynamic system can be mod
elled exactly. Control system has to deal with part to part variation, also
variation due to change in boundary conditions such as temperature, creep,
wear and tear etc. For this reason representing uncertainty and utilizing it
to achieve robust performance is one of the unique feature of robust control
theory. Linear fractional transforms also called as LFT is a powerful and
flexible approach to represent uncertainty in terms of matrices. If rq and r2
are inputs and Vi and v2 are output to a partitioned matrix then
ui = Mnr i + MX2r2
v2 M2iTi + M22v2
(4.28)
and linear fractional transform of M by A interconnect by eliminating v2
and v2
Vi [Mn + Mi2A(7 M22 A) 1M2i] 7*1
ui = Fl(M, A)ri
(4.29)
similarly v2 = Fjj(M, Au)r2 can be defined by eliminating v\ and ux.
59
4.1.3.1 Multiplicative Plant Uncertainty
Unstructured uncertainty shown in Figure (4.2) is relatively simple to define
and useful to represent unmodelled dynamics. Nominal plant P in terms of
perturbed transfer function P = (1 + AW2)P where W2 is called as weight
is a stable transfer on and A is a variable stable transfer function satisfying
11 A] [oo < 1. This uncertainty model is such that A W2 is normalized plant
perturbation away from unity. ^ 1 = AW2 and if  Al^ < 1 then,
P(ju)
P(jv)
 1
<\W2{ju>)\, Vo;
(4.30)
provides \W2(ju)  the uncertainty profile similar to disc with center 1, radius
\W\2 at each frequency point P/P. The multiplicative uncertainty can be
used effectively to account plant gain variation.
4.1.3.2 Additive Plant Uncertainty
Additive as shown in Figure (4.3) uncertainty proves useful especially when
bounds of transfer function grows with the frequency range. In particular in
sufficiently high frequencies the phase information is completely unknown.
This is a consequence of dynamic properties of which inevitably occurs in
physical systems. This gives a less structured representation of uncertainty,
for instance the statement
P(ju)) = P(ju) + Wtiju)AW2(ju), o [A(ja;)] < 1, Vw > 0, (4.31)
where W\ and W2 are stable transfer matrices characterize the frequency
structure of the uncertainty, confines the matrix P to a neighborhood of
60
the nominal model P. Disc drive standard setup with additive uncertainty
shown in Figure (4.3) where P is a set of uncertain plants where P G P
as nominal plants with K as internally stabilizing controller for P. The
complementary sensitivity and sensitivity matrix functions are defined as
usual as
50 = {I + PK)'1, T0 I So (4.32)
and
51 = (I + KPy\ To = ISi (4.33)
We assume that the model uncertainty can be represented by an additive
perturbation
P = P +
(4.34)
therefore
P(jco) P(ju) <\wadd{juj)I, Vw.
(4.35)
Additive uncertainty is most effective to define high frequency dynamic
perturbation especially when no reliable plant information is available.
4.1.3.3 Parametric Plant Uncertainty
Stability with uncertain plants can be studied using small gain theorem.
The goal is to define set of plants P that any true plant can belong. Such set
of plants can represent structured or unstructured. Structured uncertainty
61
for a plant model
P =
s2 + as + 1
with natural frequency 1 rad/sec and damping ratio o/2 could represent a
RLC circuit or springmassdamper system. If the constant is known to
the extent that it lies within [amin, amax], then structured set can be defined
as
P ) o T ^min ^ dmax 1 (4.37)
f s2 + as 41 J v '
Since most of the modal dynamics can be modelled using one or more single
degree of freedom systems such as
mx + cx + kx = u (4.38)
and the coefficients assumed to be varying within
m = fh(l + aSm), c = c(l + f38c), k = k(l + j5k) (4.39)
with 1 < dm,dc,dh < 1, represents a % uncertainty in m, represents
(3 % uncertainty in c, represents 7 % uncertainty in k. Linear Fractional
Transforms formulation of such perturbation is shown in Figure (4.4).
4.1.4 Dual Stage Servo Controller Synthesis
So far the discussion was focused on the plant dynamics and modelling un
certainty for design interconnect to synthesis Poo or fi controller presented
by Young et al. [9]. This section will focus on the weight selection and its
influence on the dual stage controller design.
62
4.1.4.1 Control Signal Penalty Weights
VCM is distinctive nonminimum phase plant and works like double inte
grator in the low frequency with lowering gain as frequency increases, also
response includes phase lag of 180 degrees. Compared to micro actuator,
VCM structural resonances occurs at significantly low frequency. Micro
actuator response looks constant at low frequency and shows low damped
resonance at high frequencies. Balancing these tradeoffs and maximize uti
lizing the strengths of both actuators is the key goal of defining the control
design weights Wvcm and WMicro The control design weights Wvcm and
Wmicto must specify any desired limitations in the control signals. The
selection of these penalty weights has significant impact on the controller
design since the two actuators have different design requirements as well
as performance capabilities. These weights can be used complementary to
enhance control loop bandwidth by utilizing low frequency capability of
VCM and broadband frequency capability of micro actuator with limited
stroke. Micro actuator has significantly faster response compared to VCM.
Minimizing the stroke requirement can be achieved by allowing high fre
quency tracking to Micro actuator and allocating low frequency work to
VCM. Figure (4.5) shows the various control weights.
While dealing with two different dynamics, one effective at low and other at
high frequency, one of the challenges is the gradual handoff of the control
outputs between VCM and micro actuator. During the gradual handoff
at intermediate frequency requires care full control over phase. Hoo or /jl
63
synthesis solution will optimally design controller for this challenge with
proper weights selection.
Since the VCM controller stroke capability is practically unlimited the
penalty for VCM is defined minimum and constant across all frequency
range, however this weight can not be zero for numerical reason
Wvcm = Â£, Vo;
(4.40)
Micro actuator is penalized heavily in the low frequency i.e. below 1kHz and
then gradually allowed to contribute more as frequency increases. However
the penalty for micro actuator is continue even in the high frequency range
to minimize the stroke thus minimizing the risk of saturating, which can
occur easily due to limited stroke availability. Various higher order weights
can be evaluated to define the frequency at which the micro actuator capa
bility can be fully kicked in:
14W = or
(s + u^A1/2)2
a
s/27t/i +1
or
;r^h V (4.4i)
^s/Vd + 2Trf2/^J
4.1.4.2 Control Signal Performance Weights
Performance weight selection includes low frequency tracking requirements
as well as high frequency disturbance rejection for Windage and other dis
turbances. Low frequency tracking requirements imply sufficient loop gain
at lower frequencies and high frequency requirements imply the peaking
64
of the error bubble after the sensitivity cross over at 0 dB. Based on the
Bodes sensitivity integral formula for stable open loop performance
where S is defined as sensitivity and L is loop gain and the relation between
those two defined as
means that the low frequency performance gain must be paid by high fre
quency amplification in the sensitivity function. Selection of sensitivity
weight has significant impact over controller low frequency integrator gain,
gain margin, phase margin and open loop bandwidth Minimizing sensitiv
ity peak, less than 2 or 6 dB can ensure the gain margin more than 4dB and
phase margin more than 30 degrees as shown in Figure (4.6). Control loop
bandwidth can be defined by sensitivity cross over which approximately
half is of the required open loop bandwidth. Also controller roll off over
high frequency, trade offs between gain and phase margin, stabilizing the
mode based on gain or phase criterion and the peaking trade offs against the
disturbance spectra are the key considerations for defining the performance
and sensitivity weights which are related by the interconnect as
(4.42)
(4.44)
guarantees satisfying %oa performance.
65
4.2
Controller Synthesis
Figure (4.5) shows the required sensitivity and performance weights used
during this synthesis exercise. VCM plant behaves like double integra
tor approximately from 100 Hz until reaches the 1st significant resonance
and rolls off by 40dB per decade. Open loop essentially shaped to 60 dB per
decade by adding integrator action in the low frequency, 20 dB per decade
in the mid frequency range by adding lead action and proper roll off based
on the high frequency dynamics. Also with dual stage %00 interconnect
the required sensitivity is defined as a combined single entity for Micro ac
tuator and VCM and the synthesis along with the control penalty weight
optimizes the distribution of the control action between Micro actuator and
VCM. With reference to the general control configuration, the standard
control problem is to find all stabilizing controllers K which minimize
II^HCfOlloo = max<7(Ji(P, K)(ju>)) (4.45)
w
The "Koo norm has several interpretations in terms of performance. One is
that it minimizes the peak of the maximum singular value of Fi(P(ju),K(ju)).
It also has a time domain interpretation as induced (worstcase) 2norm.
Let z = Fi(P, K)w then,
F(P,Jf)
IW<)I2
max T)rvrr
w(t)?o io(Â£)j
(4.46)
where j^(t)[2 = y \zi(t)\2dt is the 2norm of the vector signal. In
practice, is usually not necessary to obtain an optimal controller for the Hoo
problem, and it is often computationally and theoretically simpler to design
66
a suboptimal controller. Let 7min be the minimum value of Fj(P, P)oo
over all stabilizing controllers K such that
i)(P,4f)U<7
(4.47)
This can be solved efficiently by using the algorithm of Doyle et al. [1], and
reducing 7 iteratively, an optimum solution is approached. With assump
tions 1 6 there exists a stabilizing controller such that p(P, A')0O <7
if and only if
I.X00 > 0 is a solution to the algebraic Riccati equation [10]
ATXO0 + X^A + CfCx + X00(72P1Pf B2Bl)Xoo = 0 (4,48)
such that ReXi [A + (72PxPf B2Bj)X00\ < 0, V,; and
2Too > 0 is a solution to the algebraic Riccati equation
AtY^ + Y^At + B,BT
/ 2^1
(4.49)
such that ReXi [A + roo((72C'iC,1r CjC2))\ < 0, V< ; and
3 p{XooYinfty) ^ T
All such controllers are then given by K = Fi(Kc, Q) where
Aoo XooLoo ZooB.
Foo 0 I
c2 I 0
(4.50)
Poo = Bt2Xoo, Loo = YcoCg, ZQO = (I 'y~2Y00X00y1 (4.51)
Aqo A + 7 ^BiB^X)oo Y B2FcaZ00L00C2 (4.52)
67
and Q(s) is any stable operator transfer function such that Qoo < 7 For
Q(s) =0 we get,
K(s) = Kcn(s) = ZooL^sI A00)1F00 (4.53)
This is called a central controller and has same number of states as general
ized plant P(s) and also can be converted in to a state observer form. If the
desired controller needs to achieve ymin to within a specified tolerance, then
bisection algorithm can drive 7 until the value of 7 is sufficiently accurate.
Figure (4.8) shows the synthesis results with achieved norm 0.6577.
In this part, the Hoo synthesis interconnect with nominal VCM and Mi
cro actuator parts is developed and solved. Controller solution is used to
develop understanding of how the synthesis optimizes the control signal
distribution with highly penalized Micro actuator and low penalized VCM
plant. Synthesis results show capability of more than doubling the band
width capability with dual stage as compared to single stage controller. It
is interesting to see that VCM controller includes low frequency integrator
action carrying all workload at low frequency and continues to add more
control action in the high frequency. The VCM controller roll off was not
specified in the penalty weights mainly due to the plan for adding the uncer
tainty in the latter parts to develop a //synthesis interconnect. The Micro
actuator controller contributes minimum in the lower frequency range and
kicks in at around 1 kHz range. Note that the controller gains are with
respect to the VCM and Micro actuator plants hence cannot be compared
to oneonone basis since their plant DC gains, are different. More details
68
over the control action, Micro actuator stroke requirement will be analyzed
in the next part via a time domain simulation. The understanding for the
simulation and 'HOQ synthesis will be used to develop a complete //synthesis
with practical implementation significance.
The structured singular value // as defined by Packard [11] is a powerful
tool for analysis robust performance with a given controller. However, //
is also useful to find a controller that minimizes the given //condition is
called as //synthesis problem. At present there is no direct method to
synthesize a //optimal controller, however the more complex method for
complex perturbation is known as DGKiteration developed by Yang [12]
which combines synthesis and //analysis, often yields better results
than orderly 'Hoosynthesis.
4.2.1 Structured Singular Value //
For analyzing the stability and performance properties of multivariable
system interconnects shown in Figure (4.9) subjected to norm bounded un
certainties, structured singular value // can be used as effective tool. The few
important properties needs consideration while defining as well as control
performance are nominal stability, nominal performance, robust stability
and robust performance.
4.2.1.1 Nominal Stability
A closed loop system is stable with nominal plant that does not include any
uncertainty or perturbation, so controller K, stabilizes the system F](0, M).
69
4.2.1.2 Nominal Performance
A closed loop system satisfies the performance specifications with nomi
nal plant with no model uncertainty, so controller K, stabilizes the system
Fi(Q, M) and further satisfies the performance objective  \Fi(Fu(Q, M), K\<
1.
4.2.1.3 Robust Stability
A closed loop system is stable for all perturbed plants about the nominal
model up to the worst case model uncertainty, so if a controller K stabilizes
a system Fi(A, M), where all perturbations defined by A for a set of plants
in {P : P = FU(A, M), VA e A}.
4.2.1.4 Robust Performance
The system satisfies performance specifications for all perturbed plants
about the nominal model up to the worst case model uncertainty, so if
a controller K stabilizes a set of plant {P : P = FU(A, M), VA e A} as
well as satisfies performance objective \\Fi(Fu(A, M), K)^ < 1,VA A}.
4.2.1.5 DGK and DK Iteration
Complex fi analysis was proposed by Doyle and it involves iteration be
tween computing the scaling sets achieving complex fi upper bound and
synthesizing Hoo controller for the scaled system. For a system matrix N
70
and scaling matrix D
H{N) < mind(DiVD~1)
r>gx>
(4.54)
The starting point is the upper bound on fi in terms of scaled singular value
and goal is to find a controller that minimizes peak value over frequency of
the upper bound
by altering between minimizing .D./V'(lif).D'10o with respect to either K
or D while holding the other fixed. To start the iterations, one selects
initial stable rational transfer matrix D(s) with appropriate structure. The
identity matrix is often a good initial choice for D provided the system
has been reasonable scaled for performance. By iterating on D and G,
a controller can be designed which decreases the [x and scaled norm.
Currently, there is no explicit procedure for directly designing a controller
that optimizes jx. The /i synthesis algorithm work was extended by Young
[12] with mixed [x, the D G K iterations then proceed as follows:
(1) Estimate scaling matrices D(uS) and G{ui) for the frequency points
defined for fx analysis.
(2) Fit state space realization D(oj) and G(u) based on estimated ma
trices D(oj) and G{u>).
(3) Compute scaled system matrix M^g as a function of M and the
state space realization D(u>) and G(co) where Mdg is constructed
analogous to DMD~X.
minfmin I\DN(K)D 11 Iqq)
k KDev" v 1 11 '
(4.55)
71
(4) Compute Hoc suboptimal or optimal controller by using 7iteration
such that Fi\\Fi(MDg, < 7 for the values of 7 close to opti
mal.
(5) Find D(u) and G(u) minimizing mixed fx upper bound.
(6) If D{ui) and G(u) converges then stop the iteration, else return to
step 2.
The iterations may continue until satisfactory performance is achieved such
that Â£>./V.D_100 < 1, or until the 1Loo norm no longer decreases. One
fundamental problem with this approach is that although each of the mini
mization step (Kstep and Dstep) is convex, joint convexity not guaranteed.
Therefore, the iteration may converge to a local optimal. However, practical
experience suggests that the method works well in most cases. The order
of controller resulting from each iteration is equal to the number of states
in the generalized plant G(s) plus the number of states in the weights plus
twice the number of states in D(s).
Conceptually structured singular value is a straightforward generalization
of singular values of constant matrix. While norm is a good measure of size
of matrix, recall that the Euclidean norm was found to be a good measure
of vector length.
x = (xtx)2 (4.56)
72
While weighted Euclidian norm allowed certain components of x to be
stressed
\\Dx\ \ = (xtDtDx)z (457)
Conservatively, if  x  takes a given value, above equation provides a relative
measure of size of D. In this regard the spectral norm of a square matrix
D is defined as
 max
ll*ll=i
where Da; is defined in equation (4.58) if D and x are real. If D and x are
complex, the transpose is replaced by the Hermitian (complex conjugate)
transpose; then
INI = (xHx)> (4.59)
\\Dx\\ = (xhDhDx)* (4.60)
11Da;11 has more than one size; in fact it can take infinity values. For
dim(x) = n and dim(D) = (n.n). Ds magnitudes defined along n principle
directions by the eigenvalues of DTD or DHD. These eigenvalues are real
because DTD or DHD are symmetric or Hermitian, and the eigenvalues
positive square roots are the singular values of D. The maximum and min
imum singular values are denoted by
spectral norms of D and inverse
cr(D) = D (4.61)
\\D\\
73
1
(4.62)
a(D) =
I PI 
= min D:r
IMI=i
Hence size of D depends upon how we look at it, and it has positivereal
minimum and maximum eigenvalues.
with X can be used to determine the measure of size. The singular value
provide more reasonable measure of size.
Conservative bounds for acceptable additive and multiplicative variations
in A(s) can be based on singular values. If the original closed loop system
is A0(s)[Im + A(s)]1 is stable, then the additive perturbation <5A(s) does
not destabilize the system as long as
cr[8A(ju))\ < a[Im + A0(ju)], 0 < to < oo (4.63)
Note that the maximum singular value of the perturbation matrix is com
pared with minimum singular value of the return matrix. The comparison
must be made for all cj in [0, oo] so the singular values can be graphed in the
form of bode plots. If the test is failed at any u> stability is not guaranteed,
nevertheless, the test is conservative and stability is still possible.
To be more specific, standard feedback interconnection with stable M(s)
and A (s), one important question one might ask is how large A in the sense
of 11 Aoo can be without destabilizing the feedback system. Since the closed
loop poles are given by det(7 MA) = 0, the feedback system becomes
unstable if det{I M(s)A(s)) = 0 for some s G C+. Now let a > 0 be a
sufficiently small number such that the closed loop system is stable for all
11Aoo < a. Next increase a until amax so that the closed loop will become
74
unstable. So amax is the robust stability margin. By small gain theorem,
 = Afoo := sup a(M(s)) = sup a{M{joS)) (4.64)
OCmax sÂ£C+ w
if A is unstructured. Note that for any fixed s Â£ C+, a(M(s)) can be
written as
cr{M{s)) = 
(4.65)
mw{cr(A) : det(I M(s)A) = 0}
where A is unstructured. Or in other words, the reciprocal of the largest
singular value of M is the measure of smallest unstructured A that caused
instability of the feedback system.
To qualify the smallest destabilizing structured complex A, the concept of
singular value needs to be generalized. In view of the characterization of
the largest singular value of matrix M (s) is given by
1
Ha (M(s)) =
(4.66)
min{a(A) : det(I M(s)A) = 0}
where A is structured and the largest structured singular value of M(s)
with respect to the structured complex A. Then it is obvious that the
robust stability margin of the feedback system with structured complex
uncertainty A is
1
Oiri
sup /iA (M(s)) = sup/la (M(juj))
(4.67)
imax u
The structured singular value also some times denoted as Mu, mu or SSV.
4.2.2 Control Design via (i Synthesis
The structured singular value, (j,, is the appropriate tool for analyzing the
robustness, both stability and performance of a system subjected to struc
75
tured, LFT perturbations. In this section we cover the mechanics of a con
troller design methodology based on structured singular objective which
heavily rely on the upper bound of /i.
In order to apply the general structured singular value theory to control
design, the control system problem has been recast into the LFT setting as
shown in Figure (4.9).
The system labelled P is the open loop interconnection and contains all the
known elements including the nominal plant model and performance and
uncertainty weighting functions. The Aj,ert block is the uncertain element
from the set Apert, which parameterize all of the assumed model uncertainty
in the problem. The controller is K. Three set of input parameter enter P,
perturbation inputs w, disturbances d, and control signals u, The three sets
of outputs generated, perturbation outputs z, errors e and measurements
y
The set of systems to be controlled is described by the LFT
{Fu(P, Apert) Apert e Apert, maxa[Apert(juj)\ < 1} (4.68)
The design objective is to find a stabilizing controller K, such that for all
such perturbations Apert, the closed loop system is stable and satisfies
\\FL[Fu{P, Apert), < 1 (4.69)
'V'
perturbed plant
based on the LFT
Fl[Fu(P, Apert)K] = Fv[Fl(P, K), Apert\ (4.70)
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Therefore, the design objective is to find a nominally stabilizing controller
K such that for all Apert e Apert, maxw a[Apert(ju))] < 1 the closed loop
system is stable and satisfies
\\Fu[FL(P,K)Apert)]\\oo
Given any K, this performance objective can be checked utilizing a robust
performance test on the linear fractional transformation Fl(P,K). The
robust performance test should be computed with respect to an augmented
uncertainty structure,
A :=
Apert 0
0 Aj?
Apert ^ A.
pert
The structured singular value provides the correct test
mance that K achieves if and only if
(4.72)
for robust perfor
max.(j,&[FL(P,K)(ju)] < 1 (4.73)
b)
The goal of m synthesis is to minimize over all stabilizing controllers K,
the peak value of Ma( ) of the closed loop transfer function Fi(P, K).
More formally,
min maxfiA[FL(P,K)(ju)\ (4.74)
Kstabilizing ^
For tractability of the fj, synthesis problem,it is necessary to replace /^a( )
with the upper bound and is optimally scaled by
//a(M) < inf a(DMD~1) (4.75)
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Using the upper bound, the optimization in equation (4.75) is reformulated
as
min max min a(DwFL(P, K^u^Dj) (4.76)
Kstabilizing & DÂ£D ^
Frequency dependent function D that satisfies Dw Da for each u. The
general expression for max^
tion
min min
Kstabilizing D(s)eDAstobie ,minphase
DFl(P,K)D1
oo
(4.77)
This optimization is solved by an iterative approach, referred as DK itera
tion. To solve above equation, first D considered as fixed, stable, minimum
phase, realrational. Then solve the optimization
mm
Kstabilizing
DFl^K)!)1
(4.78)
where generalized plant P and the interconnect with D and D 1 referred
as PD, thus the optimization equation (4.78) becomes
min Ft(P0,ir)00 (4.79)
** stabilizing
Since Pd is a known step, this optimization is precisely an i?oo optimiza
tion problem. The solution to the Hoo problem is off course involves solving
algebraic Riccati in terms of a statespace model for Pd The value of
/x =1.1 for robust stability means that all the uncertainty blocks must be
decreased by a factor 1.1 in order to a guaranteed stability. Robust per
formance means that the performance objective is satisfied for all possible
plants in uncertainty set, even for the worst case plant.
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4.2.3
Synthesis Summary
D K iteration depends heavily on optimal solution for Steps 1 and 2, and
also on good fits in Step 3, preferably by a transfer function of low order.
One reason for preferring a loworder fit is that reduces the order of Hoo
problem, which usually improves the numerical properties of optimiza
tion Step 1 and also yields a controller with lower order. In some cases
the iteration converges slowly, and may be difficult to judge whether the
iterations are converging or not. If // increased instead of decreasing that
shows the numerical problems, //synthesis often results a better controller
compared to synthesis. The plant dynamics is modelled similarly for
the tfoo and //synthesis, however //synthesis performed with additive un
certainty. The weight selection explained during the H00 synthesis exercise.
Figure (4.10) and (4.11) shows synthesized controller, open loop transfer
function and sensitivity comparisons. The controller with minimum 7 and
// was selected to perform time domain simulation. Bandwidth achieved by
//synthesis is higher compared to the controller synthesized by H^ synthe
sis.
4.3 Time Domain Simulation
With development of Hoo and //synthesis, a time domain simulation model
as shown in Figure (4.12) is developed to add more capability and under
standing of specific design parameters such as closed loop analysis including
disturbance rejection. External disturbances affecting the drive track miss
79
registration. Also this simulation will be utilized to develop adaptive con
troller design for micro actuator. The model development will be performed
in Matlab and Simulink environment, however main focus will be provided
to Matlab because of its flexibility over adding adaptive control and signal
processing algorithms. Simulink environment is relatively simpler for rapid
simulation development with drag and drop and connecting the blocks sim
ilar to a wiring diagram. However, most of the Simulink blocks work like
black boxes with minimum control over the details. Developing a Matlab
simulation similar to a batch process loop, can be time consuming in the be
ginning although can be much useful for algorithm development and their
analysis. Matlab simulation developed is capable of using the controller
synthesized in the previous part without any code change and flexible and
advanced enough to reconfigure itself for any change in the controller or
plant orders. Also it is well correlated to the modelled data in frequency
domain with respect to derived transfer functions from time domain data.
4.3.1 Dual Stage Controller Simulation
Figure (4.12) shows essential blocks for simulating a dual stage controller
environments, which mainly consists of the controller with single input and
multiple outputs, VCM and Micro actuator plant. The VCM and Micro
actuator structural dynamics plant responses are modelled as a continuous
system as shown previously. Transport delay block includes the structural
and computational delay. Saturation block simulated the stroke capability
for VCM as well as micro actuator and Quantization block simulates the
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rounding off errors due to the fixedpoint precision processing capability.
The controller block includes the single input multiple output discrete time
controller.
Even though this simulation includes the essential parts the real life per
formance include much more complexities, so various capabilities to this
simulation can be added as follows Figure (4.14) shows a simulation that in
cludes Simulink SFunction usage to implement a dual stage controller with
ability to simulate fixed as well as floating math. Also the SFunction adds
flexibility for turning off the micro actuator controller or integrator suitable
to simulate various real life scenarios such as seek and settle. Adding a
SFunction is slightly complicated than using Simulink blocks since Matlab
script gets executed during each zero crossing during the simulation. Rest
of the blocks inclusive of the transfer functions for Notch Filters, VCM
and Micro actuator plant dynamics and various inputs as well as output
disturbances. Also external disturbances were simulated through RV bock
mainly for the rotational external vibration excitation. Other blocks in
clude Quantization to simulate discrete fixedpoint math, zero order hold
and time delay, and saturation blocks to simulate the stroke capability for
VCM as well as Micro actuator plants. The simulations is also capable
of computing closed loop as well as open loop performance through vari
ous switches embedded in the Simulink diagram. The simulation was well
correlated however the results are not included due increased focus on the
Matlab time domain simulation rather than Simulink for development of
more advanced adaptive controllers and also since Matlab and Simulink
81
simulations are for same results.
4.3.2 Dual Stage Matlab Batch Simulation
Adaptive algorithm development can be a complex task and requires signif
icant control over each step compared to the Simulink simulations. Batch
processing in Matlab can emulate the required control over dual stage time
domain simulations. The state space controller implementation with dual
control stated as
x(k + 1) = Ax{k) + Bu(k) (4.80)
y{k) = Cx(k) + Du(k)
where, u(k) is the measured position error signal at sample k and y(k) is
the controller command signal vector that contains VCM and micro actu
ator command signal G to R2 subspace. Floating point coefficients of state
space controller can be converted to fixed point equivalent numbers or no
tations defined by fractional binary numbers. A binary number consists of
binary point defining fractions in binary form. A fractional binary number
can be defined in following form
Nl
&iv 2) ,bi, &o]q = ^ bi2z~Q (481)
i=0
where, N is the number of bits, Q is the location of the binary point and
i are the bits in the word. The fractional binary number are often defined
as Qn numbers where n is the location of the binary point, e.g. Q14 is a
number with 14 bits after binary point. So higher the ability to define the n,
82
minimum the rounding off errors and ability is limited the DSP capability
of the processor.
Figure (4.15) shows Matlab implementation of the VCM controller, with
added flexibility to implement any order controller including real or complex
poles. Practical implementation for state space controller with fixedpoint
math capability requires proper scaling of controller coefficients in order to
minimize the rounding errors. Scaling the coefficients by power and base 2
makes the efficient usage of processor shift capabilities.
x(k + 1) = 2 sAsAx(k) + 2 sBBsBu{k) (4.82)
y(k) = 2sCCsCx(k) + 2 sDDsDu(k)
where, sa, sb, sc and sp are positive real numbers equal to the right shifts
required to represent each matrix in Qn format and ASA corresponds to
Qn representation of 2~SA and rest of the matrix respectively. The DSP
computation performed using shifts makes efficient implementation of such
algorithms.
VCM and micro actuator plants are simulated by difference equations.
xv(0 = ^_iyw(0 = 0,1,2,..., (4.83)
where q~l is a unit delay operator, and detailed Matlab implementation is
attached in the Appendix. Figure (4.15) shows the open loop simulated
and modelled transfer function estimate. Figure (4.16) shows the close
loop non repeatable run out (NRRO). In this part dual stage time domain
simulations developed and analytical models developed in this part will be
83
utilized in later parts to evaluate inverse adaptive control algorithms and
analyzing closed loop capability of dual stage actuators. The simulations
can be used to predict the micro actuator capability as well as define the
design requirement for VCM and micro actuator structural dynamics. Even
though these models and simulations are simple enough can be used as a
power full tools to predict real life drive performance in track follow mode
and combine them with another drive subsystems such as Seek and Settle.
The dual stage controllers synthesized using Hoo or //synthesis can be incor
porated easily to this simulations including ability to change the structural
dynamics or VCM and microactuator. The simulations flexible enough to
just define the controller to simulate or predict the performance with new
design.
The microactuator plant dynamics characterization is significantly different
than the VCM plants in terms of location of pole and zeros. Inversion of
nonminimum phase VCM plant often results stability concerns. However
the minimum phase Micro actuator plant is invertible and can be used
to minimize the NRRO disturbance shown in the NRRO spectra. The
Matlab batch simulation will exploit this micro actuator capability in later
parts.
84
85
Figure 4.2: Multiplicative plant uncertainty
Figure 4.3: Additive plant uncertainty
86
Magnitude (dB) Magnitude (dB)
Figure 4.4: Parametric uncertainty
25
25.5
26
26.5
27
27.5 n
10
Frequency (Hz)
Required Sensitivity
20
0
20
40
60
60
100
VCM Controller Wt
IVCM Controller Wt 
DAC Controller Wt
Performance Wt
I Performance Wt~
0.5
_iL
Frequency (Hz)
Figure 4.5: Synthesis weights
87
