1

Nonminimum Phase Dynamic Inversionfor Settle Time Applications

Brian P. Rigney, Lucy Y. Pao, and Dale A. Lawrence

Abstract—Single-track hard disk drive (HDD) seek perfor-mance is measured by settle time, ts. In this paper, we showthe effective use of feedforward dynamic inversion, coupled withreference trajectory yd generation, to achieve high performancets. Models of HDD dynamics are typically nonminimum phase(NMP), and it is well known that the exact tracking solutionfor NMP systems requires noncausal preactuation to maintainbounded internal signals. In the specific HDD operating modesof interest, anticipation of a seek command is unrealistic, andthus preactuation adds to the overall computation of settletime. Unlike many dynamic inversion tracking applications, thisnegative effect of preactuation leads to interesting trade-offsbetween preactuation delay, yd tracking accuracy, and achievablesettle performance.

We investigate multiple single-input single-output (SISO) in-version architectures, and we show that the feedforward closed-loop inverse (FFCLI) achieves superior settle performance tothe feedforward plant inverse (FFPI) in our application becauseFFCLI does not excite the closed-loop dynamics. Using the FFCLIarchitecture, we further investigate numerous NMP inversionalgorithms, including both exact inversion schemes with initialcondition preloading and stable approximate NMP inverse tech-niques. We conclude that the settle performance of the zero-order Taylor series stable NMP approximation matches the bestperformance of the exact inversion techniques in our application,and does so without the high frequency excitation required bythe Zero Magnitude Error Tracking Controller (ZMETC), or theexcessive preactuation required by the Zero Phase Error Track-ing Controller (ZPETC). Minimum energy optimal trajectorygeneration methods show that the system order n is a limitingfactor in settle performance. This confirms that the zero-orderseries method, which is capable of producing settle times in lessthan n samples, is on par with optimal approaches yet muchsimpler to implement. Multiple NMP inversion algorithms areexperimentally validated on a Servo Track Writer (STW), whichreinforces the general trends observed in ideal simulations.

Index Terms—Dynamic inversion, model inversion, nonmini-mum phase systems, settle time, hard disk drive (HDD) seeks,minimum energy trajectories.

I. INTRODUCTION

Many applications perform repetitive rest-to-rest maneuverswith unsaturated plant inputs, including single-track hard diskdrive (HDD) “seeks”. Disk drives perform repetitive single-track seeks in numerous operating modes, including servo

B. P. Rigney is an employee of Western Digital Corp. and a Ph.D. studentin the Electrical and Computer Engineering Dept., University of Colorado,Boulder, CO 80309-0425, email: brian.rigney@gmail.com. Prof.L. Y. Pao is with the Electrical and Computer Engineering Dept., Universityof Colorado, email: pao@colorado.edu. Prof. D. A. Lawrence iswith the Aerospace Engineering Sciences Dept., University of Colorado,email: dale.lawrence@colorado.edu. This work has been sup-ported through research grants from Maxtor Corporation and the NationalScience Foundation (CMS-0201459 and CMMI-0700877). Correspondingauthor: B. P. Rigney.

track writing [1]-[3], sequential data transfer [4], and man-ufacturing scans to detect media surface defects [5]. Seekperformance is measured by settle time, ts, defined as the timefrom the arrival of a seek command until the measured positionreaches and stays within an acceptable distance from the targettrack. In these operating modes, it is desirable to have eachdrive achieve its minimum ts to reduce manufacturing timeand cost, or increase data throughput. Although our motivationoriginates from disk drives, minimizing settle time is a populargoal in many diverse applications including integrated circuitmanufacturing [6][7] and scanning microscopy [8][9].

While we simplify the focus of this paper to discrete-time,single-input single-output (SISO), linear time-invariant (LTI)plant dynamics, there are still many challenges imposed bythe applications. HDD dynamics can nominally be describedby a rigid body mode with higher-order structural resonances.Over a population of plants, these resonances show structureduncertainty in lower frequency ranges, and much larger, un-structured variation at higher frequencies. Fig. 1 shows theplant magnitude distribution as a function of frequency overa population of HDDs. The distribution is tight through thedominant resonance at 6 kHz, but widens for frequenciesgreater than 7 kHz. It is extremely difficult for a low-orderparametric model to capture the complexity of the populationin this higher frequency range. The distribution also widens atextremely low frequencies due to friction nonlinearities.

A further complicating factor is the presence of nonmini-mum phase (NMP) zero dynamics. NMP dynamics can arisewhen the sensors and actuators are noncollocated [10], as inHDDs which have the magnetic reader position sensor andvoice-coil actuator on opposite ends of a flexible actuator arm.

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75% CDF

50% CDF (Median)

25% CDF

Fig. 1. Frequency magnitude response from production data over a populationof HDDs. The 25%, 50%, and 75% points of the empirical cumulativedistribution function (CDF) are shown at each frequency. This data wasprovided by Maxtor Corporation.

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Fig. 2. Block diagram of error feedback compensator C, with referenceinput r, feedforward input uff , output trajectory y, and plant dynamics P .

NMP zeros in discrete-time dynamics can also result fromfast sample rates and high relative degree [11]. As we willsee, NMP zeros require special treatment in the application ofcertain techniques for aggressive rest-to-rest maneuvers.

The computational processing power within the applicationscan be limited, particularly for HDDs where high unit volumesdictate the use of low-cost digital signal processors (DSPs).The DSPs are responsible for much more than control taskswhich further limits the amount of processing power avail-able for control. This constrains the single-track seek controldesign and necessitates the use of computationally efficientalgorithms.

Finally, the HDD error feedback compensator C is usuallydesigned for regulation purposes using knowledge of theplant dynamics P , the disturbance and noise spectra, andperformance metrics on the regulated state. Typically, thiscompensator is not ideal for aggressive single-track seekmaneuvers where settle time is the dominant performancemeasure. Further, any permanent or temporary change tothe feedback compensator for settle performance improve-ments could negatively affect regulation or require complicatedswitching to remove transients. Hence, improvements to settletime are best accomplished through exogenous inputs r anduff which enter the loop as shown in Fig. 2.

Given these complexities and constraints, we define thefeedforward settle-optimal control (FF-SOC) for single-trackHDD seeks as the exogenous input signals (r∗, u∗ff ) whichminimize ts. Due to the large uncertainty in the plant dy-namics, (r∗, u∗ff ) will vary from one unit to the next in theHDD population. The minimum achievable ts will also varyacross the population, as unmodeled high frequency dynamicsinteract with the exogenous inputs. Therefore, Iterative Learn-ing Control (ILC) [12]-[14], which iteratively computes therequired (r, uff ) such that the plant output y tracks a fixeddesired output trajectory yd, does not directly apply. If wechoose a fixed yd that every unit in the population is capable oftracking, we would be sacrificing performance on many units.Time-optimal strategies with saturated commands, as in [15]and [16], also do not directly apply. Actuator commands aremuch less than the saturation limits for single-track seeks, andsaturating actuator commands can excite unmodeled dynamicsand lengthen settle time. Initial Value Compensation (IVC)[17] uses the compensator’s initial condition response to cancelany undesirable plant output y due to the exogenous inputssignals (r, uff ). Similar to the exact inversion methods ofSection V, solving for the initial conditions requires complexcalculations relative to the application’s processing power. Fur-ther, the solution must be recomputed often as the closed-loopdynamics and exogenous inputs vary. Input-shaping [18][19]

Fig. 3. Block diagram summarizing a settle time reduction technique: param-eterize (r, uff ) with the desired output trajectory, yd, and a dynamic inverseV . Over many iterations, yd and V adapt to improve settle performance.

is a yet another technique that has been applied to reduce ts,although it is most commonly concerned with the reductionof high frequency resonance mode excitation. While the HDDplant population certainly has high frequency structural modesthat can be excited and increase ts, the lower frequency, well-damped closed-loop mode dominates the selection of (r, uff ).The use of input shaping alone is undesirable because the lowfrequency mode causes large phase lag in the input shapingfilter and lengthens ts.

Standard practices within the disk drive industry attempt todesign a fixed robust single-track seek control algorithm thatachieves an acceptable level of performance across the entirepopulation of HDDs. While this nonadaptive design is com-putationally efficient, many of the units sacrifice performancein order for the population outliers to perform adequately.Alternatively, our ultimate goal is a computationally efficientadaptive method which exploits the repetitive nature of theoperating modes to minimize ts for each unit. We havepreviously proposed [20][21] the use of adaptive dynamicinversion and adaptive reference command generation, de-picted in Fig. 3. The closed-loop system from Fig. 2 hasbeen recast as the two input, one output system G. We usea low-order LTI inverse system V and a desired commandprofile yd to parameterize the input signals (r, uff ). WhileV adapts to the structured uncertainty in the population overmany iterations, both the duration and shape of yd changeto reduce ts. Admittedly, this procedure is a suboptimalapproach to the FF-SOC problem, but, as we will show, itis capable of producing settle performance results equal totraditional optimal trajectory generation techniques [22][23],while also being computationally simpler. Other researchershave investigated the combined use of dynamic inversion andreference trajectory generation to reduce ts [7][24][25], but thecomplex off-line optimization procedures are instead aimed atrobust settle time minimization across the population and arenot amenable to on-line adaptation on low-cost DSPs.

The purpose of this paper is the selection of the inversesystem V for a settle performance objective. Although manytechniques exist for computing V [26]-[37], the best choicefor settle performance is unclear, even for nonadaptive designswithout uncertainty in G. After precisely defining ts and ourtechnique for visualizing settle performance in Section II,we apply that technique to determine the relative merits ofdifferent internal architectures and NMP inversion algorithmsfor V in Sections III-V. This analysis is performed fornonadaptive V without uncertainty in G in order to determinea performance baseline for each algorithm and architecture.In an effort to quantify the settle performance of the dynamicinversion approach, we apply minimum energy optimal state

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transfer techniques [22][23][38] and evaluate their achievablesettle performance in Section VI. This approach highlights thesystem model order as a limit of settle performance, and val-idates the capability of certain dynamic inversion techniqueswhich settle in fewer samples than the system order. Further,some dynamic inversion techniques are more computationallyefficient and amenable to future adaptive approaches. WhileSections III-VI focus on known dynamics without modelinguncertainty, Section VII investigates the effects of uncertaintywhen implementing a subset of the inversion algorithms onexperimental hardware. Section VIII summarizes the conclu-sions of the paper and describes future work combining themost promising inversion techniques for settle performancewith on-line adaptive algorithms.

II. SETTLE TIME PERFORMANCE MEASURE

Dynamic inversion algorithms are typically applied to out-put tracking problems where there is a fixed desired outputtrajectory yd [26]-[36]. Frequency responses of the closed-loopdynamics between the reference trajectory yd and the outputtrajectory y are used to compare different inversion techniquesand quantify measures of the tracking error [31][33]-[36]. Forsettle performance, tracking a particular yd is not the objective.Further, it is difficult to relate frequency responses to theachievable settle performance. In this section, we explicitlydefine our measure of settle time, present a computationallyefficient parameterization of a family of yd trajectories, and usea graphical technique to view settle performance as a functionof yd.

A. Settle Time Measurement

Settle time ts is defined as the time from the arrival of aseek command until the measured position reaches and stayswithin the settle boundary surrounding the desired set-point,as shown in Fig. 4. The settle time, as expressed in numberof samples, is ks, where ts = ksT and T is the sample time.In HDD single-track seek applications, there is no penalty onovershoot or undershoot in the plant output y; settle time isthe only performance measure of interest, with typical settleboundaries set at ±5% of a track. Further, we do not considerany constraints on plant input magnitude or spectral content.High frequency excitation is of no concern in our applicationunless it lengthens settle time.1

Preactuation, a change in the plant input before the seekcommand arrives, and postactuation, a change in the plantinput after the seek has settled, are common side-effects ofusing dynamic inversion algorithms. The example seek inFig. 4 clearly shows both artifacts, although the postactuationis truncated to save space. As we will discuss in Section V,both preactuation and postactuation are ultimately a functionof the zero dynamics of the system to be inverted. Preactuationrequires anticipation of the incoming seek command, which isdifficult to synchronize in our HDD operating modes because

1In reality, the acoustical signature of HDD seeks is a concern. While wedo not treat this constraint in our work, it would be possible to combine aweighted frequency-domain measure of the plant input u with ts to form amodified settle performance cost function.

Fig. 4. Example input and output trajectories for a single-track seek, withpreactuation, postactuation, and total settle time clearly noted.

of sizeable non-deterministic software and hardware delays.Therefore, ts must be delayed by the preactuation time tp inorder to accommodate the preactuation sequence. Similar tosettle time, the preactuation time tp, expressed in number ofsamples, is kp, where tp = kpT . The partitioning of kp intok0 and p samples will be fully explained in Section V. Thets definition can now be explicitly stated as the time fromthe initial change in (r, uff ) until y reaches and stays withinthe settle boundary. Unlike many dynamic inversion trackingapplications, this negative effect of preactuation leads tointeresting trade-offs between preactuation delay, yd trackingaccuracy, and achievable settle performance. These will beimportant factors in the analysis of NMP inversion algorithmsin Section V.

Postactuation is of less concern for all the HDD operatingmodes of interest. Servo track writing, manufacturing defectscans, and sequential data transfer typically dwell at each trackset-point for at least one revolution of the disk. This dwelltime is then dependent on the spin speed of the disk, but canbe a factor of 10 greater than the settle time. As long as thepostactuation is complete within this dwell period, it will notaffect performance. As illustrated in Section V, this is the casein our HDD applications.

B. yd generation

As previously mentioned, our solution procedure to improvesettle performance for each unit in the population combinesadaptive dynamic inversion and adaptive yd trajectory gener-ation. We therefore need a parameterization of yd trajectoriesthat is computationally simple but able to produce a wide rangeof yd’s, from unaggressive to extremely aggressive. [24] and[25] have previously proposed using a combination of mono-tonically increasing polynomials, parameterized by a scalarvalue corresponding to the time duration of the trajectory.Given the limited processing power in the HDD applications,we are encouraged to find an even simpler parameterization.

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Fig. 5. Desired output trajectory yd is generated from a bang-bangacceleration pulse. The duration and aggressiveness of yd is parameterizedby d.

Motivated by the solution to the time optimal controlproblem for a rigid body, we use a family of yd trajectoriesgenerated from the double integral of a bang-bang accelerationpulse, as shown in Fig. 5. The bang-bang acceleration signalhas the following Z-transform:

Accel(z) =1

d2T 2

(z2d − 2zd + 1)z2d − z2d−1

, (1)

=(z − 1)d2T 2

(d−1∑

i=0

zi

)2

z2d−1, (2)

where T is the sample time and 2d is the total duration ofboth the Accel signal and yd. The Z-transform of yd can thenbe written as

YD(z) =T 2

2(z + 1)(z − 1)2

Accel(z) , (3)

=1

2d2

(z + 1)(z − 1)

(d−1∑

i=0

zi

)2

z2d−1. (4)

This simple parameterization provides a single scalar value, d,that can select the aggressiveness of the seek, while also beingextremely computationally efficient. Although future adaptivesolutions will change d on-line for each unit in the population,this paper uses a set of predetermined values for d, yieldingyd durations from an unaggressive 6.80 ms (100 samples atthe application sample rate T ) to a very aggressive 0.136ms (2 samples) single-track HDD seek. We then evaluate thesettle performance of different architectures and algorithmsfor V over this family of yd’s. In the final section of thispaper, we discuss the potential need for increased-complexityyd parameterizations for further settle time reduction.

C. Actual Versus Desired Settle Time

The minimum achievable ts over a family of yd’s is a usefulscalar performance measure for comparing different designsfor the system V , but conceals some interesting details. Muchcan be learned about the individual inversion algorithms andarchitectures by observing how the settle performance changesas a function of yd. We therefore use a graphical visualization

Fig. 6. Actual settle time ts versus desired settle time td provides agraphical measure of settle performance as yd varies from unaggressive tovery aggressive trajectories.

of settle performance by plotting the actual settle time, ts, asa function of the settle time, td, of the desired yd trajectory.Fig. 6 illustrates the expected behavior of the actual versusdesired settle time. Ideal performance corresponds to ts = tdfor any choice of yd, although this is unrealistic for plants withhigh frequency modeling errors. For aggressive yd’s with shorttd’s, there is increased plant input energy at high frequencywhich can excite unmodeled dynamics and increase ts. Thiscan cause the minimum achievable settle time on particularunits to occur at less than the most aggressive yd. The effectof preactuation on settle performance is also evident in thefigure. Preactuation increases ts by tp, relative to td, and canlimit the amount of settle time reduction possible. While tpcould be a function of the specific yd trajectory, it is shown asa constant offset in Fig. 6. Fixed preactuation delay is typicalfor the algorithms we investigate in Section V.

III. MODEL INVERSION ARCHITECTURES

We are interested in determining the single-input, two-output dynamic inverse V from Fig. 3. This is challengingbecause many multiple-input, multiple-output (MIMO) inver-sion schemes require the number of inputs to equal the numberof outputs [26]-[28][33]-[36]. Fortunately, constraining theinternal architecture of V can convert the original problem intoSISO inversion. In this section, we investigate two commonfeedforward architectures for V which result in SISO systemsfor inversion: the plant inverse (FFPI) architecture and theclosed-loop inverse (FFCLI) architecture. We then show howto modify the FFPI architecture to match the FFCLI input-output relationship, which we refer to as the feedforwardmodified plant inverse (FFMPI). Exact models for the plant areassumed in order to understand the performance implicationsof the inverse architecture alone.

A. Feedforward Plant Inverse (FFPI) Architecture

The plant inverse feedforward architecture with error feed-back, as in Fig. 7, is routinely used in industry and has beeninvestigated by many in the literature [26]-[29][39]. The model

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Fig. 7. Block diagram of plant inverse (FFPI) architecture.

inverse V in Fig. 3 that implements this architecture is[

uff

r

]= VFFPI yd , (5)

where VFFPI =[

P−1

z−kp

], (6)

and the delay z−kp and plant inverse P−1 are further explainedbelow.

In order to algebraically describe the dynamics from yd to yin Fig. 7, it is helpful to first partition the plant into minimumphase (MP) and nonminimum phase (NMP) zero polynomials

P (z) =Bm(z)Bn(z)

A(z), (7)

where Bm is the numerator polynomial including all plant MPzeros, Bn contains all plant NMP zeros, and A is the plantdenominator polynomial. We can then express the plant inverseP−1 as

P−1(z) =A(z)B−1

n (z)zkpBm(z)

. (8)

We use the variable B−1n to denote the inverse of the NMP

zero polynomial Bn. The ˜ modifier on both P−1 and B−1n

signifies that these quantities are not necessarily the exactinverses of P and Bn, respectively. When P is NMP, B−1

n

can take on numerous forms, including Hurwitz polynomialapproximations of the exact inverse, and both stable andunstable rational polynomial transfer functions. Moreover,kp samples of delay have been added to P−1 to maintaincausality. A noncausal inverse can result from a strictly properplant transfer function P with nonzero relative degree, orthrough the use of noncausal algorithms for B−1

n . As notedin Section II, anticipation of a seek start command would berequired to implement a noncausal inverse, which is unrealisticin the HDD operating modes of interest. We therefore mustadd kp samples of delay to P−1 and also to the input r, asin Fig. 7. The various NMP inversion algorithms differentiatethemselves in the way they compute B−1

n and the kp samplesof delay required. For the purposes of this section, we remainvague about the specific form of B−1

n and value for tp;Section V goes into more detail on the different NMP inversionalgorithms.

Given these definitions, the FFPI system dynamics frominput yd to output y are

y

yd

∣∣∣∣FFPI

=

(B−1

n BnCDA + CNBmBn

)

zkp (CDA + CNBmBn),

=

(CDAB−1

n + CNBm

zkpCNBm

)HCL . (9)

CN and CD are the Hurwitz numerator and denominatorpolynomials of the compensator C, which is assumed stable,exactly proper, and minimum phase. HCL is the closed-looptransfer function between r and y

HCL =CNBmBn

CDA + CNBmBn. (10)

We have dropped the z argument from the various polynomialsand B−1

n for convenience. When P is MP, Bn = 1, B−1n = 1

and (9) simplifies to z−kp . In this case, the plant output yexactly tracks the desired output yd with kp samples of delay,which we refer to as delayed exact tracking. This is also truein the NMP case when B−1

n is taken as the exact (unstable)inverse of Bn, although (9) is no longer internally stable.We will address this issue in Section V. When P−1 is anapproximation to the exact inverse, (9) shows that the closed-loop dynamics are present in the y response. This can havedrastic effects on ts because the closed-loop natural responseis often relatively slow to settle, having limited bandwidth dueto stability robustness concerns in the presence of unmodeleddynamics.

B. Feedforward Closed-Loop Inverse (FFCLI) Architecture

The closed-loop inverse (FFCLI) architecture, pictured inFig. 8, is also common in the literature [31]-[36]. Referring toFig. 3, the dynamic inverse V which implements the FFCLIarchitecture is [

uff

r

]= VFFCLI yd , (11)

where VFFCLI =[

0H−1

CL

]. (12)

Here, H−1CL is the inverse of the closed-loop system from r to

y:

H−1CL =

ACL

zkpBmCL

B−1n , (13)

with

ACL = CDA + CNBmBn , (14)BmCL

= CNBm . (15)

The plant and closed-loop systems share the same NMP zerosand relative degree because the compensator C is assumed tobe MP and exactly proper. Applying this inverse as in Fig. 8,we find the FFCLI dynamics from input yd to output y are

y

yd

∣∣∣∣FFCLI

=BnB−1

n

zkp. (16)

Similar to the minimum phase FFPI case, when P , andthus HCL, are MP, (16) simplifies to pure delay and weachieve delayed exact tracking. This also holds, along with

Fig. 8. Block diagram of closed-loop inverse (FFCLI) architecture.

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the internally unstable caveat, when HCL is NMP and H−1CL is

the exact unstable inverse. The differences between FFPI andFFCLI appear when H−1

CL is an approximate inverse. In thiscase, (16) shows that the closed-loop dynamics are no longerexcited. Further, if B−1

n is a polynomial approximation, (16)has a finite impulse response (FIR). This can have significantsettle performance advantages when the dominant closed-loopdynamics are slow relative to the settle performance objective.Fig. 9 shows a simple simulation comparison between FFPIand FFCLI for a single aggressive yd input trajectory. Thissimulated response was generated using a Zero Phase ErrorTracking Controller (ZPETC) [33] approximation for B−1

n ,which we will discuss in Section V. We present Fig. 9 here tosimply highlight the settle performance advantages of FFCLI.The FFPI response has a 0.27 ms longer ts due to theexcitation of the closed-loop dynamics, consistent with (9).After a fixed number of samples, the FIR FFCLI responsehas exactly reached the set-point. The delay between yd andy for both architectures is a result of the delays added to theinverse system to maintain causality. We will further explainthis effect in Section V as well.

The settle performance differences between FFPI and FF-CLI can also be a strong function of uncertainty in the plantdynamics, which has not been addressed in this section. [21]shows that the settle time advantages of FFCLI over FFPIdiminish for certain types of plant modeling error. Section VIIIwill discuss future efforts with adaptive inverse systems, wherethe goal of adaptation is to reduce the effects of plant modelingerror on settle performance. It is therefore useful to quantifythe best possible settle performance without uncertainty foreach architecture, and is the rationale behind the exclusion ofuncertainty in our analysis.

C. Feedforward Modified Plant Inverse (FFMPI) Architecture

The FFPI architecture suffers reduced settle performancedue to excitation of the closed-loop dynamics. This excitationoccurs because the tracking error r− y is nonzero when P−1

is not the exact plant inverse. We can augment the FFPIarchitecture with the yd prefilter M , as in Fig. 10 and [37],to remove the closed-loop dynamics from the y response. Werefer to this architecture as the modified plant inverse (FFMPI).

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

Time [ms]

Po

siti

on

[tr

acks

]

Settle Boundary

yd

FFPIFFCLI

Fig. 9. Simulated comparison between FFPI and FFCLI for a singleaggressive yd using a ZPETC approximate inverse.

Fig. 10. Block diagram of modified plant inverse (FFMPI) architecture.

The dynamic inverse V in Fig. 3 which implements FFMPI is[

uff

r

]= VFFMPI yd , (17)

where VFFMPI =[

P−1

M

]. (18)

The FFMPI dynamics from input yd to output y are

y

yd

∣∣∣∣FFMPI

=

(B−1

n BnCDA + zkpMCNBmBn

)

zkp (CDA + CNBmBn).

(19)

We can see from (19) that the FFMPI dynamics from yd to ycan match the FFCLI dynamics by selecting M as

M =B−1

n Bn

zkp. (20)

This selection for M can also be viewed as a modificationof the tracking error equation to e = Myd − y such that eremains zero and the closed-loop dynamics are not excited.

While the yd to y dynamics are the same for FFMPI andFFCLI, the r and uff inputs to the closed-loop system aredifferent. In FFCLI, uff is zero and r is the output of thesystem H−1

CL in (13), which has poles equal to the rootsof BmCL . If the compensator is designed to provide phaselead, CN can have roots near the unit circle [40]. This willcause H−1

CL to have a slow response and potentially longpostactuation after ts. With FFMPI, r is the output of thesystem M which, depending on the form of B−1

n , can haverelatively little postactuation. The P−1 system’s output is uff ,whose response is governed by the roots of Bm, as in (8). Theroots of CN do not factor into the postactuation calculationfor the closed-loop inputs in the FFMPI architecture. Thiscan have advantages in applications that cannot tolerate longpostactuation.

For the balance of this paper, we will use the FFCLIarchitecture for all simulation and experimental results. Wechoose this architecture because:

1) FFCLI achieves the same settle performance as FFMPI,and better settle performance than FFPI, for our appli-cation. The slow closed-loop dynamics are not presentin the output y.

2) The postactuation benefits of FFMPI are of no concernin the targeted HDD operating modes.

3) FFCLI only requires the implementation of a single filterwhich has computational benefits.

4) FFMPI requires an accurate model of the plant dynam-ics, which is more difficult to experimentally identifythan an accurate model of the closed-loop dynamics inour application.

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B]

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ase

[deg

]

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Model

(a) HCL Frequency Response

0 1 2 3 40

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Po

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(b) HCL Step Response

Fig. 11. Experimental and modeled frequency and step responses of STW closed-loop system.

Before investigating the settle performance of numerous NMPinversion algorithms in Section V, we next discuss the closed-loop dynamic model identification procedure used to generatethe results in this paper.

IV. CLOSED-LOOP EXPERIMENTAL SYSTEMIDENTIFICATION

Having chosen the FFCLI architecture for the system V ,we now need a model for HCL. This model is requiredboth for the model-based NMP dynamic inversion algorithmsand for a simulation study of their settle performance. Weuse a Servo Track Writer (STW) [2], provided by MaxtorCorporation, as our experimental testbed and focus the iden-tification procedure on the STW’s closed-loop dynamics. TheServo Track Writer is used to magnetically encode the initialservo position information on the magnetic media during HDDmanufacturing. The STW has its own voice-coil motor (VCM)and precision encoder that mechanically interface with theHDD actuator arm and HDD VCM through an opening inthe HDD baseplate. Traditionally, the STW moves the HDDactuator one track, magnetically encodes one revolution ofservo position information, and repeats this process until alltracks are written. Modern HDDs can require the STW to makeover 500,000 single-track seeks per disk drive. The single-track seek distance is determined by the HDD track density;we use a single-track step size of 1 µrad as a representativeangular track width for a modern HDD. The STW has anencoder sensor resolution of 0.5 nanorad, and a compensatorsample time of T = 68 µs.

We experimentally identify the STW closed-loop dynamicsby injecting a pseudo-random sinusoidal sequence as thereference input r in Fig. 2. The input sequence’s sinusoidalamplitudes are designed to give adequate signal-to-noise ratioover a wide frequency band of interest, and the frequencies areselected such that each sinusoid has an integer number of peri-ods within the sequence. Fig. 11(a) shows the experimentallyidentified frequency response of HCL, with a weighted least-squares model fit to the frequency data. The frequency-domainweighting vector and model order were selected through trial-and-error to produce a reasonable matching of the experimen-tal step response, shown in Fig. 11(b). The weighted least

squares model for HCL is

HCL(z) =0.10988(z + 0.4947)

(z − 0.1541)(z2 − 1.859z + 0.8695)

· (z2 − 1.874z + 0.8807)

(z2 − 1.24z + 0.4409)

· (z2 + 2.389z + 1.574)

(z2 + 1.233z + 0.878), (21)

which matches both the experimental frequency and stepresponses well. This 7th-order model has a closed-loop band-width near 1 kHz, unity DC gain, a relative degree of 2, ahigh frequency structural mode near 5.3 kHz, and 2 NMPzeros outside the unit circle at z = − 1.1946 ± j0.3838.

The step response with 5% settle boundaries in Fig. 11(b)also highlights requirements for improved settle performance.The ts for the step response of the HCL model is 1.6 ms.While there is some evidence of small high frequency oscil-lations in the measured output, the step response settle timeis dominated by the lower frequency closed-loop dynamics.Any enhancement to settle performance over the populationof HDDs must account for these lower frequency dynamics toproduce settle times much shorter than 1.6 ms.

V. NONMINIMUM PHASE DYNAMIC INVERSIONALGORITHMS

Dynamic inversion is complicated by the presence of NMPzeros, even for LTI SISO systems. The NMP zeros of theoriginal system become unstable poles in the inverse system.For the HDD application using the FFCLI architecture, thiscauses the system in Fig. 8 to exactly track yd (with possibledelay) while r grows unbounded. Many techniques exist tocompute exact and approximate solutions for the inverse dy-namics, H−1

CL, with bounded input signal r. We have organizedthe algorithms considered in this paper with the hierarchicalchart in Fig. 12. There is a major division in NMP algorithmsbetween those that stably approximate the exact unstableinverse of HCL, and those that use the exact unstable inversedirectly. We first investigate the exact inversion procedure andreview techniques that use the initial condition responses ofH−1

CL and HCL to maintain bounded r while exactly trackingyd with delay. We then turn our attention to the most applicableapproximation methods for the HDD application.

8

Fig. 12. Hierarchical chart of the NMP inversion algorithms discussed inthis paper.

A. Exact NMP Inversion for SISO LTI Systems

The causal exact inverse of HCL is

H−1CL(z) =

ACL(z)B−1n (z)

zkpBmCL(z)

=ACL(z)

zpBmCL(z)Bn(z)

, (22)

where p is the relative degree of HCL. We have explicitlychosen to separate the delay from the denominator poly-nomials in order to focus the analysis on the contributingfactors to preactuation delay. In this case, kp = p samples ofpreactuation due to the requirement that H−1

CL be causal. Usingthis exact inverse in a FFCLI architecture produces delayedexact tracking

Y (z) = HCL(z)H−1CL(z)Yd(z) =

1zp

N(z)D(z)

, (23)

where the Z-transform of yd has been written as the ratio oftwo polynomials, N and D. As previously noted, the systemin (23) is internally unstable when HCL is NMP, and ther input signal grows without bound. In some applications,extremely short yd signals or NMP zeros close to the unitcircle may result in r inputs that are effectively boundedand useful over a short interval. (Method 1 in Fig. 12).When truly bounded r is required over any interval, as inour targeted applications, many exact inversion algorithmsuse the initial condition responses of H−1

CL and HCL at k0

(referred to in Fig. 4) to maintain bounded r while stillachieving delayed exact tracking [24]-[30]. The authors of[29] give explicit continuous-time formulas for computing therequired initial conditions, and [26]-[28][30] derive additionalpreactuation sequences of length k0 that preload the requiredinitial conditions of HCL. In this case, the total preactuation iskp = k0+p. Similar in spirit to [30], we review a discrete-timepolynomial development of the required initial conditions,simplified for SISO LTI system dynamics.

We begin by writing the Z-transform of the closed-loop in-

put r, including both the forced and initial condition responses

R(z) =ACL(z)

zpBmCL(z)Bn(z)

N(z)D(z)

+IC(z)

zpBn(z)BmCL(z)

. (24)

Here, IC is the numerator polynomial of the initial con-dition response of H−1

CL, with the ˜ modifier again usedto refer to the inverse system. By choosing to describeH−1

CL with an observable canonical state-space realization(Aobs , bobs , cobs , dobs ), the initial condition responsenumerator polynomial simplifies to

IC(z) = z cobs adj(zI − Aobs )x(k0) , (25)=

[zn zn−1 · · · z

]x(k0) , (26)

where n is the order of ACL and the identity used in (26)can be found in [38, p. 660]. In order to compute the initialstate vector x(k0) required to cancel the unstable modes inthe forced response, we must isolate Bn in (24)

R(z) =U1(z)

zpBmCL(z)D(z)+

U2(z)Bn(z)

+IC(z)

zpBn(z)BmCL(z). (27)

The polynomials U1 and U2 result from the partial fractionexpansion of the forced response and must satisfy

U1(z)Bn(z) + U2(z)zpBmCL(z)D(z) =

ACL(z)N(z) . (28)

Hence, if x(k0) satisfies the following equation[

zn zn−1 · · · z]

x(k0) =zpBmCL(z) (γBn(z)− U2(z)) , (29)

where γ is an unknown scalar constant, then the r response isbounded and has the following Z-transform

R(z) =U1(z)

zpBmCL(z)D(z)

+ γ . (30)

We postpone discussion of the existence of x(k0) and thecomputation of γ in (29) until we develop the required initialconditions on HCL.

An unfortunate consequence of removing the unstable com-ponents from r is that we no longer achieve delayed exacttracking. The Z-transform of the output y, again includingboth forced and initial condition responses, is

Y (z) =Bn(z)

ACL(z)U1(z)

zpD(z)+ γ

BmCL(z)Bn(z)

ACL(z)

+IC(z)

ACL(z), (31)

where IC(z) is the numerator polynomial of the initialcondition response of HCL. Similar to the inverse dynam-ics, an observable canonical state-space realization for HCL,(Aobs, bobs, cobs, dobs), leads to

IC(z) =[

zn zn−1 · · · z]

x(k0) , (32)

9

with x(k0) representing the initial condition vector of HCL.By isolating the delayed exact tracking portion of the forcedresponse in (31)

Y (z) =N(z)

zpD(z)− U2(z)BmCL

(z)ACL(z)

+γBmCL

(z)Bn(z)ACL(z)

+IC(z)

ACL(z), (33)

we can use x(k0) to cancel the unwanted components[

zn zn−1 · · · z]

x(k0) =BmCL

(z) (−γBn(z) + U2(z)) . (34)

Equations (29) and (34) have solutions x(k0) and x(k0),respectively, if

1) The order of the right-hand side (RHS) of each equationis less than or equal to n. By definition, the order ofzpBmCL

Bn is n and the order of zpBmCLU2 is less than

or equal to n. Therefore the order constraint is easilysatisfied.

2) The RHS of each equation has at least one root at z = 0.If BmCL

does not have a root at z = 0, then γ mustsatisfy

γ =U2(0)Bn(0)

, (35)

where Bn(0) is nonzero by definition.The unstable mode cancellation in the inverse system can be

analytically computed and exactly enforced in the feedforwardcontroller software, i.e., by only implementing (30); preciselysetting the initial conditions of the physical closed-loop HCL

system is much more difficult. We discuss four techniquesfrom Fig. 12 to address this difficulty.

Method 2 - HCL State Transfer to Desired x(k0):As a byproduct of discrete-time controllability, there exists asequence r(k) that transfers the state of HCL from 0 to x(k0)in at most n samples [38, Sec. 2.3]. While it may be possibleto control the state of HCL from 0 to x(k0) in ` samples,where ` < n, this requires the very restrictive condition thatx(k0) lies in the range space of the `-step input-to-state map.For our HCL and the required x(k0)’s over the set of yd’s, thisdoes not apply and we are left with kp = n + p = 9 samplesof preactuation for the example system (21) in this paper. Thismethod of preloading initial conditions is not typically usedin standard dynamic inversion tracking applications because yis non-zero during preactuation and causes increased trackingerrors. In contrast, it is a perfectly legitimate technique in thissettle time application where preactuation is already includedin ts. Unfortunately, as we will see, 9 samples of preactuationis a very long delay compared to other inversion techniques.

Method 3 - Infinite Preactuation: The authors of [26]and [28] derive a continuous-time noncausal input r(t) overthe preactuation interval t ∈ (−∞, t0] which drives the state ofHCL from rest to the required x(t0) while maintaining y(t) =0. A similar procedure is detailed in [30] for a noncausal inputr(k) in discrete-time, which we briefly review here. Consider(27), without the initial condition response of H−1

CL included

R(z) =U1(z)

zpBmCL(z)D(z)+

U2(z)Bn(z)

. (36)

In order to compute a bounded r(k) from R(z), we use thestandard right-sided inverse Z-transform for the stable firstterm in (36) and a left-sided inverse Z-transform for theunstable second term [41]. The left-sided inverse transformfor U2(z)

Bn(z) has a region of convergence (ROC) that includesthe unit circle and produces a bounded noncausal sequencewhich is nonzero on k ∈ (−∞, k0 − 1]. This noncausalportion of r(k) drives the state of HCL to the required initialconditions x(k0) such that the undesired forced response termsin (33) are cancelled and y achieves delayed exact tracking.A portion of the r(k) preactuation sequence for the examplesystem was previously shown in Fig. 4. An infinite kp causesan obviously unrealistic infinite ts, and we thus investigatetruncation strategies.

Method 4 - Truncated Preactuation: The example sys-tem’s r(k) in Fig. 4 converges quickly to zero as k → −∞, al-lowing for practical implementation through truncation. Whiletruncated preactuation is no longer an exact inversion ap-proach, it is directly motivated by the exact inversion develop-ment and we thus include it with the other exact algorithms.The authors of [27] and [30] discuss truncation strategies whenyd is only known over a finite window ahead of the currentsample. Instead, we take advantage of full knowledge of yd inour application, which allows for the partition in (36). Wethen use the first k0 samples of the left-sided inverse Z-transform of the unstable second term in (36) as the truncatedpreactuation sequence applied on k ∈ [0, k0 − 1]. Largevalues for k0, and consequently tp, result in small errors inx(k0), very little tracking error (disregarding delay), but largeoffsets in the calculation of ts. There is thus an interestingtrade-off between the length of tp and the achievable settleperformance, explored further in Fig. 13. Unlike delayedtracking performance, which generally improves with increas-ing noncausal preactuation, settle performance is negativelyaffected by increasing tp. Surprisingly, a zero-length left-sidedinverse transform sequence (k0 = 0 and tp = 2) provides thebest settle performance. This is the equivalent of leaving theHCL initial conditions zero, and is further explored in theMethod 5 discussion.

Method 5 - Preload x(k0) while Leaving x(k0) = 0:Setting k0 = 0 is equivalent to ignoring the unstable secondterm in (36) and leaving x(k0) = 0. The output y doesnot exactly track yd with delay because the other forcedresponse terms in (33) are present in the output. The trackingerror is dependent on the size of the undesired terms inthe y response, which in turn is determined by the degreeto which the closed-loop dynamics HCL are excited by thedesired reference trajectory yd. If the undesired modes aresmall enough in amplitude, good settle performance may stillbe obtained. Fig. 14 shows the settle performance of thistechnique compared to the performance of the Method 2 statetransfer technique to preload x(k0) for our example system.The actual settle performance ts as a function of the desiredsettle time td for Method 2 in Fig. 14(a) clearly demonstratesthe negative effect of the kp = 9 sample preactuation topreload x(k0). Preloading x(k0) achieves a much slowerminimum settle time (0.680 ms) than Method 5 (0.408 ms)which leaves the initial conditions of the physical system at

10

Fig. 13. Minimum achievable settle time ts versus preactuation timetp for exact inversion with truncated preactuation and noncausal Taylorseries approximation. The minimum achievable settle time for other inversiontechniques with fixed tp are also shown.

zero.2 We also notice a drastic increase in the Method 5 settletime for the most aggressive yd, which we further explorewith the y output responses in Fig 14(b). While the Method 2trajectory perfectly matches a (very) delayed version of yd,there is a large amount of both high frequency resonance andslow closed-loop mode excitation in the Method 5 response.The extra terms in (33) cause tracking errors that grow asyd becomes more aggressive, ultimating yielding an outputresponse with a longer settle time. Referring back to Fig. 13,we know that increasing k0 in an attempt to approximatelypreload x(k0) will only result in longer settle times. Thebenefits of reducing the response from the extra terms in (33)are overshadowed by the added preactuation delay, ultimatelyincreasing ts.

B. Approximate NMP Inversion for SISO LTI Systems

We now present three methods for stably approximatingB−1

n : the Zero Phase Error Tracking Controller (ZPETC,Method 6 in Fig. 12), the Zero Magnitude Error TrackingController (ZMETC, Method 7), and noncausal Taylor se-ries approximation (Method 8). While there are other stableinversion techniques, these three are common in practiceand the literature, and also perform best in our application.After discussing these three algorithms and comparing theirperformance to the exact inversion results, we conclude thediscussion of NMP inversion algorithms with a summarizingtable. Included in the table is another approximate inversiontechnique, H∞ model matching [31][42] (Method 10), that weomit in the discussion because of space constraints and imple-mentation drawbacks. [43] shows that the results presented inthis section are highly dependent on the location of the NMPzeros.

Method 6 - Zero Phase Error Tracking Controller(ZPETC): The ZPETC dynamic inversion method was intro-duced in [33], and has since become ubiquitous in the output

2The uneven td horizontal spacing in Fig. 14(a) is an artifact of the discreteyd sample locations in relation to the settle boundary, and is present in allsimulation and experimental results.

0 0.5 1 1.5 20

0.5

1

1.5

2

td [ms]

t s [m

s]

Ideal

HCL

State Transfer

x(k0)=0

(a) Actual settle time (ts) vs. desired settle time (td)

0 0.5 1 1.50.8

0.85

0.9

0.95

1

1.05

1.1

1.15

Time [ms]

Po

siti

on

[tr

acks

]

Settle Boundary

(b) Output y responses for most aggressive yd

Fig. 14. Settle performance comparison between HCL state transfertechnique to preload x(k0) and leaving x(k0) = 0.

tracking literature [31][32][34]-[36]. The method is based oncanceling the phase shift (but not the gain change) due tothe NMP zeros by reflecting those zeros about the unit circle.We review the algorithm by first starting with the NMP zeropolynomial

Bn(z) = bmnzmn + bmn−1zmn−1 + · · ·+ b0 , (37)

where mn is the number of NMP zeros. The zero phase errorapproximate inverse to Bn is

B−1n (z)

∣∣∣ZPETC

=b0z

mn + b1zmn−1 + · · ·+ bmn

Bn(1)2, (38)

which yields the following FFCLI transfer function from yd

to y

Y (z)Yd(z)

∣∣∣∣ZPETC

=Bn(z) B−1

n (z)∣∣∣ZPETC

zkp

=Bn(z)

(b0z

mn + b1zmn−1 + · · ·+ bmn

)

zp+2mnBn(1)2. (39)

Bn(1)2 has been used in the approximation in (38) to maintainunity DC gain in (39). While (39) is FIR, which has desirablesettle performance qualities, we have also added kp = p+2mn

samples of delay to maintain causality of H−1CL. This ultimately

limits the settle performance of ZPETC.Method 7 - Zero Magnitude Error Tracking Con-

troller (ZMETC): Although not referred to by this name,the ZMETC dynamic inversion method has been studied in

11

both [31] and [37]. In contrast to ZPETC, ZMETC cancelsthe magnitude response (but not the phase shift) due to theNMP zeros by reflecting those zeros about the unit circleand converting them to poles in the inverse system. Againbeginning with the Bn polynomial in (37), the zero magnitudeerror approximate inverse to Bn is

B−1n (z)

∣∣∣ZMETC

=1

b0zmn + b1zmn−1 + · · ·+ bmn

, (40)

which yields the following FFCLI transfer function from yd

to y

Y (z)Yd(z)

∣∣∣∣ZMETC

=Bn(z) B−1

n (z)∣∣∣ZMETC

zkp

=Bn(z)

zp (b0zmn + b1zmn−1 + · · ·+ bmn)

. (41)

Unlike ZPETC, the ZMETC system only requires kp = psamples of delay to maintain causality, which has settle timebenefits. Unfortunately, those benefits can be negated by thepresence of pole dynamics due to the zero magnitude errorapproximation. Depending on the distance of the NMP zerosaway from the unit circle, those poles can be excited by yd

and cause excursions in y outside the settle boundary.Method 8 - Noncausal Taylor Series Approximation: The

noncausal Taylor series approximation seeks to approximate1

Bnwith a noncausal polynomial

B−1n (z)

∣∣∣Taylor

=

mT∑

i=0

αizi

Bn(1)mT∑

i=0

αi

, (42)

where mT is the order of the series approximation and the αi

sequence is derived from the Taylor series expansion of 1Bn

,as in [34]. The resulting FFCLI transfer function from yd toy is

Y (z)Yd(z)

∣∣∣∣Taylor

=Bn(z) B−1

n (z)∣∣∣Taylor

zkp

=

Bn(z)mT∑

i=0

αizi

zp+mT +mnBn(1)mT∑

i=0

αi

. (43)

While (43) is FIR, kp = p + mn + mT can be large forhigh-order series approximations. Similar to the trade-offsbetween settle performance and tracking accuracy with theMethod 4 truncated preactuation, increasing mT improves theapproximation to 1

Bnand yields better (delayed) tracking ac-

curacy, while deteriorating the achievable settle performance.Referring back to Fig. 13, we plot the minimum achievable tsover a range of series orders, from 0 to 6. This yields a kp

range from 4 to 10 for our system in (21) due to the extra delayin (43) from the relative degree and NMP polynomial order.Again similar to the Method 4 truncated preactuation, theshortest series, a zero-order approximation, provides the best

settle performance. The tracking improvements that come fromhigher-order approximations are negated by the accompanyingdelay in the calculation of ts. We further explore the zero-orderseries approximation in the Method 9 discussion.

Method 9 - Zero-Order Series Approximation: Thesimplest stable approximation technique is to ignore the NMPzeros in HCL and assume Bn = 1. This is the equivalentof a zero-order Taylor series approximation, and leads to thefollowing form for B−1

n

B−1n (z)

∣∣∣Zero−Order

=1

Bn(1), (44)

which yields this FFCLI transfer function from yd to y

Y (z)Yd(z)

∣∣∣∣Zero−Order

=Bn(z) B−1

n (z)∣∣∣Zero−Order

zkp

=Bn(z)

zp+mnBn(1). (45)

As before, the term Bn(1) maintains unity DC gain of (45).Unlike ZMETC, the zero-order series preserves the desirableFIR response in (45), while using less preactuation delay thanZPETC. We now use simulation results on the example systemto compare the settle performance of ZPETC, ZMETC, and azero-order series approximation.

Fig. 15(a) shows the actual settle performance as a functionof the desired settle time for these three stable approximateinversion schemes on the example system (21). The effectsof the preactuation delay are immediately clear. ZPETC haskp = 6 samples of preactuation, the zero-order series haskp = 4, and ZMETC has kp = 2. This results in a higherminimum achievable settle time for ZPETC (0.408 ms) thanfor ZMETC (0.340 ms) or the zero-order series (0.340 ms).Even though ZMETC has less preactuation delay than thezero-order series, it does not achieve a lower minimum settletime. Fig. 15(b) explains why this occurs by examining the youtput responses for the most aggressive yd. While the zero-order series and ZPETC produce FIR responses, the ZMETCtrajectory shows an increase in ts due to large oscillationsin y. The extra poles that have been inserted as part of theZMETC approximation in (41) have a high natural frequencynear 6.6 kHz. They are increasingly excited as yd becomesmore aggressive until they cause y to initially exceed the settleboundary limits. While the closed-loop compensator could betuned to limit the high frequency oscillation of a particularmode, this type of response is of general concern consideringthe application’s high frequency unstructured uncertainty. Fur-ther, even after ts, the y trajectory is close to exceeding thesettle boundary. Disturbances and sensor noise on y can leadto very inconsistent ts’s for a given yd.

To further illustrate the differences between the inversionalgorithms considered here, including both exact and ap-proximate methods, we plot the closed-loop input r for themost aggressive yd in Fig. 16. The presence of the extrapoles in the ZMETC approximation cause extremely largehigh frequency excitation in r. The exact inversion schemewith x(k0) = 0 also has large high frequency excitation inr. In contrast, ZPETC and the zero-order series use much

12

0 0.5 1 1.5 20

0.5

1

1.5

2

td [ms]

t s [m

s]

IdealZPETCZMETCZero−Order Series

(a) Actual settle time (ts) vs. desired settle time (td)

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

Time [ms]

Po

siti

on

[tr

acks

]

Settle Boundary

(b) Output y responses for most aggressive yd

Fig. 15. Settle performance comparison between ZPETC, ZMETC, and the zero-order series stable approximate inversion algorithms.

TABLE ICOMPARISON OF SETTLE PERFORMANCE ATTRIBUTES FOR DYNAMIC INVERSION ALGORITHMS

Algorithms Min ts [ms] Preactuation kp [samples] Order of H−1CL Implementation Concerns

Exact:HCL state transfer to x(k0) 0.680 n + p = 9 n = 7 Computationally complexx(k0) = 0 0.408 p = 2 n = 7 Computationally complex

Stable Approximation:ZPETC 0.408 p + 2mn = 6 n + mn = 9 NoneZMETC 0.340 p = 2 n = 7 Sensitivity to high frequency uncertaintyZero-Order Series 0.340 p + mn = 4 n = 7 NoneH∞ Model Matching 0.340 0 13 Very computationally complex

lower amplitude inputs to the closed-loop without large highfrequency components. Ultimately, the lack of high frequencyexcitation makes the ZPETC and zero-order series more appro-priate for our application with high frequency uncertainty. Alsoevident in Fig. 16 is the postactuation required by the NMPinversion algorithms. Although the plot does not show the fullpostactuation period, all of the r trajectories converge to 1 by3.0 ms. Given that our HDD operating modes need to wait 1full revolution of the magnetic media between seeks, and 1revolution roughly lasts between 5 and 15 ms, this amount ofpostactuation is of no concern.

Table I concludes the discussion of NMP inversion algo-rithms by summarizing the best settle performance, preactua-tion required, computational complexity, and implementationconcerns of each algorithm discussed. The minimum achiev-

Fig. 16. Comparison of closed-loop input r computed by exact inversion withx(k0) = 0 and the ZPETC, ZMETC, and zero-order series stable approximateinversion algorithms.

able ts is also graphically depicted in Fig. 13. Within theexact algorithms, transferring the state to the required x(k0) isunrealistic because it requires too much preactuation. Leavingthe HCL initial conditions x(k0) equal to zero produces goodsettle performance, but requires complicated computations topartition the systems as in (27), and ultimately limits itsusefulness in future adaptive applications on our HDD testbed.The ZMETC stable approximation method produces a betterminimum settle time than the x(k0) = 0 method, but doesso with increased high frequency excitation. ZPETC and thezero-order series both have smooth closed-loop inputs withoutthe high frequency excitation, but ZPETC requires too muchpreactuation delay and its settle performance suffers. The H∞model matching technique for deriving H−1

CL has not beendiscussed, but a summary of the pertinent settle performancedata is included in the table. This method produces inversesystems that achieve high performance settle times, but re-quires larger dimensional H−1

CL filters. Further, the synthesisprocedure would be difficult to make adaptive on our limitedcomputational hardware. The zero-order series stable approxi-mation technique stands out as the method which provides thebest settle performance in our simulation environment, whilenot requiring computationally difficult filter implementationsor large high frequency energy in the input r to the closed-loop.

VI. OPTIMAL STATE TRANSFER COMPARISON

Table I shows that multiple NMP inversion algorithmsachieve 5 sample (0.340 ms) settle times in simulation. Par-ticularly, the zero-order series algorithm accomplishes thiswith a comparatively smooth input r to HCL. In order to

13

further evaluate the benefits of such an algorithm, we consideranother settle performance benchmark for HDD single-trackseeks. As an attempt to provide this benchmark, we applyan alternative, and arguably more natural, approach to settleperformance through the use of optimal control algorithms. Wereview a simple method to generate minimum energy optimaltrajectories in this section, and use this analysis to highlightcertain settle performance limitations of this technique.

We begin the discussion of minimum energy inputs bydefining the `-step input-to-state map Lr: R`×1 7→ Rn×1 [38,Sec. 9.2]

Lr =[

(Aobs)`−1

bobs (Aobs)`−2

bobs · · ·Aobsbobs bobs

], (46)

where we continue to use the observable canonical realizationfor HCL from Section V. As the name suggests, Lr is usefulin expressing the influence of the input r on the state of HCL

at sample `

x(`)− (Aobs)`x(0) = Lrr

= Lr

[r(0) r(1) · · · r(`− 1)

]T. (47)

Given x(0), x(`), and Lr, the minimum ‖r‖2 solution to (47)for ` ≥ n is

` ≥ n : r =

LrT

(LrLr

T)−1 [

x(`)− (Aobs)`x(0)

]. (48)

LrLrT is the controllability grammian, Wc, which is invert-

ible if ` ≥ n and (Aobs, bobs) is controllable. When ` < n,it is likely there is no exact solution for r which transfers thestate from x(0) to x(`). In this case, the r which minimizes∥∥∥Lrr−

[x(`)− (Aobs)

`x(0)

]∥∥∥2

is

` < n : r =(Lr

T Lr

)−1

LrT

[x(`)− (Aobs)

`x(0)

].

(49)When x(`) − (Aobs)

`x(0) lies in the range space of Lr, an

exact solution exists and can be computed through (49).By setting ` = td and allowing td to vary over the same

range as in Section V, we can use (48) and (49) to com-pute minimum energy inputs that transfer HCL’s state fromx(0) = 0 to x(`). This allows a straightforward comparisonwith the settle performance results from the dynamic inversiontechniques, but requires knowledge of the final state x(`).We take the simplistic approach of requiring HCL to be atsteady-state after ` (i.e., no postactuation), which results inthe following computation for the final state

x(`) = (I −Aobs)−1

bobs rss . (50)

Here, rss is the steady-state input and (I−Aobs) is invertiblebecause ACL is Hurwitz. Given that the DC gain of HCL is1, rss is simply the desired HDD single-track set-point. When` < n, we do not in general perfectly achieve this x(`) andresort to simulation analysis on our example system (21) tostudy this effect.

Fig. 17(a) shows the settle performance over the fixed tdrange. For all ` ≥ 7, y achieves ideal settle performance,

but there is a drastic increase in settle time for ` < 7. Theminimum achievable settle time is 0.476 ms, two samplesslower than several inversion techniques including the zero-order series. The output trajectories in Fig. 17(b), zoomed inon the settle region, provide an explanation for the increase insettle time. While the ` = 7 trajectory perfectly reaches andstays at x(`), the ` = 6 trajectory has some residual error inthe final state which excites the slow closed-loop dynamics.As we have previously seen, excitation of the closed-loopmode drastically increases settle time. The system order thusconstrains the minimum achievable settle time using minimumenergy optimal trajectories in this application. Further, whilenot viewable in Fig. 17(b), the output trajectories before settleare extremely large, with overshoot 40 times the size of theset-point. We would expect these trajectories to be extremelysensitive to high frequency modeling errors.

This minimum energy optimal state transfer analysis isoverly restrictive relative to the inversion algorithms we havestudied. We constrain the optimal trajectories to use no postac-tuation and require that they achieve the desired set-pointexactly on the `th-sample. All of the inversion algorithmsuse postactuation, and some further take advantage of thesettle boundary region by causing excursions in y that donot increase ts. The authors of [8], [22] and [23] studythe optimal state transfer problem for rest-to-rest motions inmuch more detail. They develop techniques to select differentx(0) and x(`) states, along with corresponding preactuationand postactuation trajectories, which yield lower total inputenergy and smoother trajectories in general. Even in theirmore complex analysis of the optimal state transfer problem,the system order n still constrains the minimum achievablesettle time. We therefore view the inversion algorithms thatachieve n − 2, or 5, sample settle times as being very highperformance, on par with optimal techniques. They also havethe extra benefits of being simpler computationally than theoptimal state transfer algorithms, and are more amenable tofuture on-line adaptation on low-cost DSPs.

VII. EXPERIMENTAL IMPLEMENTATION

Based on simulation analysis of modeled STW hardware,we have shown that the ZMETC and zero-order series stableapproximate inversion techniques produce high performancets’s. We now seek to validate that conclusion by implementingthose algorithms on the STW experimental testbed, while alsoincluding ZPETC experimental results for comparison. Due tononrepetitive noise and disturbances on the measured positionsignal, we average over 100 single track seeks for each yd.Fig. 18(a) shows the average ts as a function of td for thethree algorithms of interest. Similar to the simulation data, theeffects of preactuation are evident for the slower td’s, althoughthe delay is greater than predicted by simulation for all threealgorithms. For the most aggressive yd’s, all three algorithmssee an increase in the average ts. The minimum achievableaverage settle time is 0.490 ms for ZMETC, 0.645 ms forZPETC, and 0.573 ms for the zero-order series. Along with anincrease in average ts for the more aggressive yd’s, Fig. 18(b)shows all three algorithms also have increased variability in

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(a) Actual settle time (ts) vs. desired settle time (td) (b) Output y responses for ` = 6 and ` = 7 trajectories

Fig. 17. Settle performance for the minimum energy optimal state transfer trajectories.

0 0.5 1 1.5 20

0.5

1

1.5

2

td [ms]

mea

n(t

s) [m

s]

IdealZPETCZMETCZero−Order Series

(a) Average settle time (ts) vs. desired settle time (td)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

td [ms]

std

(ts)

[ms]

(b) Std. of settle time (ts) vs. desired settle time (td)

Fig. 18. Experimental settle performance comparison between ZPETC, ZMETC, and the zero-order series stable approximate inversion algorithms. 100 tssamples per td were collected and both average and standard deviation data are presented.

ts by plotting the standard deviation over 100 seeks at eachtd.

In all three cases, the experimentally determined minimumaverage ts is higher than the simulated minimum ts in Table I.We further investigate the cause of this behavior by plottinga sampling of the most aggressive output responses for eachof the three algorithms in Fig. 19. All three trajectories showincreased undershoot and phase lag relative to the simulationresults in Section V. This is indicative of lower frequencymodeling errors in the identified model for HCL, ratherthan excitation of high frequency resonance dynamics. Theundershoot is present in all output responses, even for theless aggressive yd trajectories, and it is responsible for theincreased ts relative to simulation for the slower yd’s. Conceiv-ably, these lower frequency parametric errors in HCL couldbe adaptively identified on-line to further improve the settleperformance in Fig. 18(a). The ZMETC algorithm does showmore high frequency excitation, which is consistent with thesimulation results. This also explains the increased variabilityin the ZMETC ts distribution in Fig. 18(b) for the mostaggressive yd. The excursions in the y response are so close tothe upper settle boundary that each individual seek could have

widely varying ts. It would be much more difficult to removethis behavior with on-line adaptive algorithms because HCL

is not well known at high frequencies.

As a final experimental investigation, we increase the hard-ware loop gain by 15% and re-test the zero-order series NMPalgorithm. A 15% change in the loop gain is common within apopulation of HDDs due to actuator torque constant variability,among many other factors. It is also common on a singleSTW over time because the encoder sensor gain changes as afunction of angular position. Fig. 20(a) compares the previouszero-order series experimental data with data taken on thegain-increased hardware. There is an obvious deterioration insettle performance when the HCL model does not correctlymatch the hardware. Similar to the previous experimentalresults, the dominant closed-loop mode excitation is the causeof the increased ts, demonstrated in Fig. 20(b). Without on-line adaptive algorithms to remove the effects of parametricuncertainty similar to this, we do not expect the use of dynamicinversion algorithms to yield improved settle performanceacross the population of HDDs.

15

0 0.5 1 1.5 20

0.5

1

1.5

2

td [ms]

mea

n(t

s) [m

s]

Ideal0% Gain Increase15% Gain Increase

(a) Average settle time (ts) vs. desired settle time (td)

0 0.5 1 1.5 20.8

0.85

0.9

0.95

1

1.05

1.1

1.15

y [t

rack

s]

Time [ms]

Settle Boundary

(b) Average experimental output trajectories for most aggres-sive yd

Fig. 20. Experimental settle performance comparison using the zero-order series NMP inversion algorithm on hardware with an without artificial gainincrease.

VIII. CONCLUSIONS AND FUTURE WORK

Dynamic inversion, coupled with reference trajectory gener-ation, is an effective and practical means to achieve high per-formance settle times. Unlike many NMP dynamic inversiontracking applications, which freely use noncausal preactuationto minimize the yd − y tracking error, the HDD single-trackseek applications of interest cannot anticipate the start of aseek command. Therefore preactuation adds to the overallsettle time and leads to interesting trade-offs between theamount of preactuation delay, yd tracking accuracy, and theachievable settle performance. In this paper, we review thedevelopment of common dynamic inversion architectures andalgorithms for tracking applications, and evaluate them withinthis settle performance framework.

Through algebraic comparisons, we study 3 different SISOfeedforward inversion architectures: plant inverse (FFPI),closed-loop inverse (FFCLI), and modified plant inverse(FFMPI). In settle time applications, especially when theclosed-loop compensator is necessarily low-bandwidth, weshow FFCLI is a superior architecture. The FFPI architecturewill excite the dominant closed-loop dynamics and can drasti-cally lengthen settle time. We provide a simulation example ofthis effect on our identified STW system model, and provide atechnique to modify the FFPI implementation, called FFMPI,to recover the FFCLI settle performance.

0 0.5 1 1.5 20.9

0.95

1

1.05

1.1

Time [ms]

y [t

rack

s]

Settle Boundary

ZMETCZPETCZero−Order Series

Fig. 19. Experimental output trajectories for the most aggressive yd.

Using FFCLI, we then investigate numerous NMP inversionalgorithms in settle time applications. The algorithms exam-ined include both exact dynamic inversion with initial condi-tion preloading and stable approximate NMP inversion. Theexact inversion approaches have clear advantages in trackingapplications, but require significant amounts of preactuation tosetup initial conditions. This negatively affects ts, especiallywhen the desired settle performance is on the same orderas the required preactuation. We show that on our examplesystem, it is surprisingly advantageous to use no preactuationand leave the closed-loop initial conditions zero. While thisapproach leaves unwanted forced response terms in the closed-loop output y, their response is small and does not substantiallyincrease ts.

In contrast to the computationally complex exact inversiontechniques, we show that three stable approximate inversionalgorithms are capable of achieving similar levels of settleperformance, with much simpler implementations. Particu-larly, the settle performance of the zero-order Taylor seriesapproximation exceeds the best performance of the exactinversion techniques, and does so without the high frequencyexcitation required by ZMETC, or the excessive preactuationrequired by ZPETC. In this case, the yd to y dynamics areFIR, irregardless of the location of the NMP zeros. While azero-order series approximation would never be considered asa performance tracking solution, the low-order FIR dynamicsbetween yd and y make it an extremely desirable settleperformance algorithm. In a simulation of our example system,the minimum achievable settle time using a zero-order seriesapproximation is less than the order of the closed-loop model.Minimum energy optimal trajectory generation methods showthat the system order is a limiting factor in settle performance,and confirm that this approximate inversion method is on parwith optimal approaches, yet much simpler to implement.

The NMP inversion architecture and algorithm analysis doesnot include modeling uncertainty in an effort to provide abaseline performance analysis. In order to validate the conclu-sions of this study, we show experimental results that support

16

the major trends seen in simulation. In general, all algorithmstested achieve slightly slower experimental ts than simulationdue to low frequency modeling errors in the inverse system.Moreover, when applying the dynamic inversion algorithms toexperimental hardware with a 15% increase in the loop gain,settle performance deteriorates, again due to lower frequencyclosed-loop mode excitation. A 15% error in the loop gainis one type of common parametric uncertainty in the plantpopulation, and supports the need for adaptive solutions thatcan compensate for lower frequency parametric errors.

Future work will focus on adding on-line adaptation to thesedynamic inverse algorithms using repetitive information. Asconfirmed by our experimental results, plant modeling errors,especially at lower frequencies, can cause a significant reduc-tion in settle performance. We would like to recover the idealsimulation performance over the population of plant dynamicsthrough an adaptive inverse system focused on lower andmiddle frequency parametric uncertainty. Unfortunately, thedynamic inverse will always be a reduced-order approximationto the true closed-loop dynamics because of high frequencyunmodeled modes. When aggressive yd trajectories begin toexcite these unmodeled modes, it may be possible to usethe repetitive information to adaptively shape yd using morecomplex parameterizations and improve settle performancefurther.

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[43] J. Butterworth, L. Y. Pao, and D. Abramovitch. “The Effect ofNonminimum-Phase Zero Locations on the Performance of FeedforwardModel-Inverse Control Techniques in Discrete-Time Systems,” to bepublished in Proc. of the American Control Conf., 2008.

Brian P. Rigney Brian P. Rigney received a B.S.in Aerospace Engineering from the University ofNotre Dame in 1998 and a M.S. in Aerospace Engi-neering from the University of Colorado at Boulderin 2000. He worked for Maxtor Corporation from2000-2006, specializing in Servo Track Write andSelf-Servowrite technologies. He is currently a Ph.D.candidate in Electrical Engineering at the Universityof Colorado at Boulder, and an employee of WesternDigital Corporation in the Self-Servowrite division.His interests include precision nano-scale motion

control and the application of advanced control concepts to computationally-constrained embedded systems.

Lucy Y. Pao Lucy Y. Pao received the B.S., M.S.,and Ph.D. degrees in Electrical Engineering fromStanford University, and she is currently a Professorin the Electrical and Computer Engineering Depart-ment at the University of Colorado at Boulder. Shehas interests in the areas of control systems (with ap-plications to flexible structures, atomic force micro-scopes, disk drives, tape systems, power converters,and wind turbines), multisensor data fusion (with ap-plications to unmanned autonomous vehicles, satel-lites, and automotive active safety systems), and

haptic and multimodal visual/haptic/audio interfaces (with applications toscientific visualization and spatial communication). She is currently an IEEEControl Systems Society (CSS) Distinguished Lecturer (2008-2011), a 2008Visiting Miller Professor (University of California at Berkeley), a memberof the 2008 International Federation of Automatic Control (IFAC) TriennialWorld Congress Young Author Prize Selection Committee, and General Chairfor the 2013 American Control Conference.

Dale A. Lawrence Dale A. Lawrence is an As-sociate Professor in the Department of AerospaceEngineering Sciences at the University of Coloradoat Boulder. He obtained a B.S. in Electrical Engi-neering from Colorado State University in 1980, anda M.S. and Ph.D. in Electrical Engineering fromCornell University in 1982 and 1985, respectively.His current research interests are in the areas of diskdrive control, dynamics and control of small space-craft and unmanned aerial vehicles, and design andcontrol of haptic interfaces and micromanipulators.