Robust Control Synthesis Techniques for Multirate and Multisensing ...

Ryozo Nagamune Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada e-mail: [email protected]

Xinghui Huang Seagate Technology, 389 Disc Drive, Longmont, CO 80503 e-mail: [email protected]

Roberto Horowitz Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740 e-mail: [email protected]

1

Robust Control Synthesis Techniques for Multirate and Multisensing Track-Following Servo Systems in Hard Disk Drives This paper proposes controller design methods, specially for track-following control of the magnetic read/write head in a hard disk drive (HDD). The servo system to be considered is a general dual-stage multisensing system, which encompasses most of the track-following configurations encountered in the HDD industry, including the traditional single-stage system. For the general system, a robust track-following problem is formulated as a time-varying version of the robust H2 synthesis problem. Both dynamic and real parametric uncertainties, which are typical model uncertainties in trackfollowing control, are taken into account in the formulation. Three optimal robust controller design techniques with different robustness guarantees are applied to solve the synthesis problem. These are mixed H2 / H⬁, mixed H2 / ␮, and robust H2 syntheses. Advantages and disadvantages of each method are presented. Multirate control, which is inherent to control problems in HDDs, is obtained by reducing multirate problems into linear time-invariant ones, for which there are many useful theories and algorithms available. Most of the techniques proposed in this paper heavily rely on efficient numerical tools for solving linear matrix inequalities. 关DOI: 10.1115/1.4000835兴

Introduction

Track-following control of the magnetic read/write head in hard disk drives 共HDDs兲 is of great importance in meeting recent and future requirements of extremely high track density. For a given system consisting of several components such as a suspension, sensors, and actuators, servo control should achieve optimal trackfollowing performance to meet several objectives. Optimal control will not only realize small track-misregistration but also give us useful information regarding the limitations of a given system, as well as useful information on how to modify the system structure. In addition to optimality, robustness is essential in trackfollowing control. This is because there are many disturbances affecting the control system, such as measurement noise, track runout, windage, and external shock. Moreover, a controller has to be designed so that it maintains acceptable track-following performance for hundreds of thousands of HDD units with slightly different dynamics. This paper proposes several robust and optimal control methods for a general servo system, called the dual-stage multisensing 共DSMS兲 system. The DSMS system has two actuators and several sensor measurements, and is expected to be necessary for achieving the highly precise track-following that will be required in future HDDs. For this multivariable control system, it is not easy to systematically design a controller that provides both optimality and robustness by using classical control theory. Therefore, we will apply advanced robust control theories such as mixed H2 / H⬁, mixed H2 / ␮, and robust H2 syntheses, to the present multivariable control problem. In order to enhance performance, we should exploit the freedom of using different sampling/hold rates in the DSMS system. Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 28, 2005; final manuscript received June 30, 2009; published online February 2, 2010. Assoc. Editor: Yossi Chait.

In HDDs, the sampling rate of the position error signal 共PES兲 is determined by the disk spinning speed and the number of servo sectors, while the sampling/hold rates of other sensors, such as a sensor measuring the head relative to the suspension tip, or a vibration sensor in the suspension, are flexible. It is natural to presume that an increase in these rates will improve trackfollowing performance. In this paper, we will assume arbitrary sampling/hold rates. The paper is organized as follows: In Sec. 2, a multirate robust track-following problem is formulated mathematically. Sec. 3 reviews the method in Refs. 关1,2兴 for the reduction in multirate control problems to time-invariant control ones. Section 4 presents three robust control design methods that solve the formulated robust track-following problem approximately. Section 5 gives a simple example for the synthesis of a dual-stage trackfollowing controller using the robust H2 synthesis technique. The linear matrix inequalities 共LMIs兲 used in this paper are presented in the Appendixes A and B. Even though this paper assumes a general structure for a dualstage multisensing system, it plays a role in theoretically supporting the multirate, multi-input multi-output 共MIMO兲 controller synthesis techniques used in Ref. 关3兴, which presents a comparison of sequential single-input single-output 共SISO兲 and MIMO track-following control synthesis techniques for dual-stage and multisensing servo systems in hard disk drives. Readers are referred to Ref. 关3兴 for more background on the track-following control of dual-stage servo systems for HDDs and for further simulation results of the controller synthesis techniques presented in this paper.

2

A Robust Track-Following Control Problem

In this section, we will formulate a multirate robust trackfollowing control problem to be tackled in this paper. The formulation is general enough to cover most of the track-following control problems encountered in the magnetic disk drive industry,

Journal of Dynamic Systems, Measurement, and Control Copyright © 2010 by ASME

MARCH 2010, Vol. 132 / 021005-1

Downloaded 02 Feb 2010 to 169.229.32.138. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

∆ z

w ∆

∆

w

G e n e r a liz e d p la n t

2

y S

K

2

u H

Fig. 1 A generalized plant with an uncertainty block ⌬ and a multirate controller HKS

such as single- and dual-stage controls, irrespective of the type of the secondary actuator, and the locations/number of sensors. Practical examples of track-following control, which reduces to the formulation given below, are presented in Sec. 5, as well as in Ref. 关3兴. Let us consider a discrete-time1 linear time-invariant generalized plant with an uncertainty block 共see Fig. 1兲

冤冥冤 z⌬

A

B⌬

B2

Bu

C⌬ D⌬⌬ D⌬2 D⌬u

z2 = C2 D2⌬ D22 D2u y Cy Dy⌬ Dy2 0

冥冤冥 w⌬ w2

共1兲

u

w⌬ = ⌬z⌬

共2兲

where we have used the standard notation:

冋册 A B

C D

共3兲

ª D + C共zI − A兲−1B

Here, u is the input vector of length 2, which consists of signals to the voice coil motor 共VCM兲 and an auxiliary mini- or microactuator, y is the measurement vector 共of any length兲, typically consisting of the PES, the suspension vibration signal measured by PZT sensors, the position of the magnetic head relative to the gimbal, as measured by a microactuator relative position sensor, and so on, z2 is the control output vector, typically consisting of the PES and input amplitudes, and w2 is the disturbance vector of all undesirable signals, such as the track runout, windage, and measurement noise. All the matrices in Eq. 共1兲 are constant and assumed to have compatible dimensions. The generalized plant is comprised of the VCM and secondary actuator dynamics, as well as weighting functions. The uncertainty block ⌬ is assumed to be a diagonal matrix in the set Bª

冦

⌬ ª diag关␦1, . . . , ␦ p,⌬V,⌬ M 兴

␦ j 苸 BR, j = 1, . . . ,p

⌬V 苸 BH⬁, ⌬ M 苸 BH⬁

冧

共4兲

where p is the number of parametric uncertainties, and BR ª 兵r 苸 R:兩r兩 ⱕ 1其 BH⬁ ª 兵f 苸 H⬁:储f储⬁ ⱕ 1其 The real uncertainty ␦ j is interpreted as a parameter variation in the dynamics of the VCM and the auxiliary actuator, such as gain, damping ratio, and resonance frequency. Dynamic uncertainties ⌬V and ⌬ M are typically due to high-frequency unmodeled dy1 Throughout this paper, we assume that, if a plant model is originally given in continuous-time, it has been discretized with the fastest sampling/hold rate.

021005-2 / Vol. 132, MARCH 2010

namics in the VCM and the secondary microactuator, respectively. Remark 1. It may happen that some parametric uncertainties appear repeatedly as ␦ jI. However, since the subsequent discussions are almost unchanged even in such cases, we just consider the case of nonrepeated parametric uncertainties. Denote the operator from w2 to z2 by Tz2w2. This operator depends on the uncertainty ⌬ and a multirate controller HKS, where S and H mean a multirate sampler and a multirate hold, respectively. Thus, we show the dependence explicitly as Tz2w2共HKS , ⌬兲. Note that the operator Tz2w2共HKS , ⌬兲 is time-varying in general due to the multirate sampler and hold. Then, a multirate robust track-following control problem can be formulated as follows. PROBLEM 1. For given multirate sampler S and hold H with fixed sampling and hold rates, design a controller K that stabilizes exponentially the closed-loop system for all ⌬ 苸 B, and minimizes the worst-case rms value of z2 against Gaussian white noise w2, or equivalently, solve the optimization problem min max储Tz2w2共HKS,⌬兲储2

K苸K共B兲 ⌬苸B

共5兲

where K共B兲 is the set of all controllers that exponentially stabilize the closed-loop system for all ⌬ 苸 B, and 储 · 储2 denotes the ᐉ2 seminorm defined for time-varying systems in p. 73 of Ref. [4]. This is a multirate robust performance synthesis problem with parametric and dynamic uncertainties. We remark that this problem is general in that it contains, as special cases, single-stage, single-sensing, as well as single-rate cases. Unfortunately, the formulated problem is difficult to solve exactly with existing control theory and computational tools, because of the nonconvexity and relatively large size of the typical track-following control problem. Therefore, in Sec. 4, we will present design methods to solve this problem in certain approximate cases.

3

Multirate Control

Before proceeding with the exposition of control design techniques for the formulated robust performance synthesis problem, in this section, we will review a way to transform a multirate control problem into a time-invariant one, for which there are many useful theories and numerical algorithms available. To this end, we follow the techniques used in Refs. 关2,1,4兴. For the ease of notation, only in this section, we remove the uncertainty block ⌬, as well as the signals z⌬ and w⌬, and consider a simplified generalized plant

冋册 z2 y

冤

A

B2

Bu

= C2 D22 D2u Cy Dy2 0

冥冋册 w2 u

共6兲

However, even with the uncertainty block ⌬ and corresponding channels w⌬ and z⌬, the argument in this section remains analogous. 3.1 Reduction in Time-Varying Control. First, by combining the generalized plant with the multirate sampler and hold, we will obtain a periodic time-varying system 共see Fig. 2兲. The explicit form of such a system will be derived next. A multirate sampler S is expressed mathematically as ˜ 共k兲 = ⌫共k兲y共k兲, S:y

k = 0,1,2, . . .

共7兲

where ⌫共k兲 is a diagonal matrix with diagonal entries of 0 or 1. If the ith measurement is sampled at time k, the 共i , i兲-entry of ⌫共k兲 is set to 1; otherwise, it is set to 0. In the single-rate case, ⌫共k兲 = I for any k = 0 , 1 , 2 , . . .. We assume that the sampler is periodic with a period Ts, i.e. ⌫共k + Ts兲 = ⌫共k兲,

k = 0,1,2, . . .

共8兲

Transactions of the ASME


冤冥冤

A p e r io d ic t im e - v a r y in g s y s t e m

2

S

~y

˜x共k + 1兲

w


y

H u

z2共k兲

2

冋册 us

冋册

where the subscripts s and f stand for slower and fastest respectively, and channels with slower sampling rates are gathered at the top of the vector u without loss of generality. Here, the term “fastest rate” refers to the fastest rate among not only hold rates, but also sampling rates. Thus, if the fastest sampling rate is faster than any hold rate u consists of only us. Then, the hold is a mapping

冋册冋册 ˜us

→

us

共10兲

uf

which can be described as a dynamic linear time-varying system

冤冋册冥冋 xh共k + 1兲 us共k兲

=

u f 共k兲

Ah共k兲 Bh共k兲 Ch共k兲 Dh共k兲

Ch共k兲 ª

冋

Ins − ⍀共k兲 0

册

,

册冤冋册冥 xh共k兲

˜us共k兲

冋

⍀共k兲 0 0

In f

共15兲

˜u共k兲

0

共11兲

册

冋

册

˜B 共k兲 ª BuDh共k兲 u Bh共k兲

册

˜ 共k兲 ª D , D ˜ 共k兲 ª D D 共k兲 D 22 22 2u 2u h ˜ 共k兲 ª ⌫共k兲D D y2 y2

共16兲

Note that system 共15兲 includes all the information about the multirate sampler and hold. 3.2 Reduction in Time-Invariant Control. Next, we will apply known controller design methods for time-varying systems to system 共15兲, thereby obtaining a multirate controller. It is proven in Refs. 关4,1,2兴 that many important control synthesis problems for time-varying systems can be solved in a very similar way to those for time-invariant systems. Moreover, for periodic time-varying systems, these problems can be reduced to finite-dimensional convex optimization problems, which can be solved by using efficient numerical techniques for LMIs. To be more concrete, using the matrices in the periodic timevarying plant 共Eq. 共15兲兲 with its period T, let us consider an auxiliary linear time-invariant system

˜u f 共k兲

冋册 z2 y

Bh共k兲 ª 关⍀共k兲,0兴

Dh共k兲 ª

w2共k兲

˜ 共k兲 ª ⌫共k兲关Cy 0 兴 C y

where xh is the state vector of H, and for k = 0 , 1 , 2 , . . . Ah共k兲 ª Ins − ⍀共k兲,

冥冤冥 ˜x共k兲

˜ 共k兲 ª 关C2 D2uCh共k兲兴 C 2

共9兲

uf

冋

˜B 共k兲 ª B2 , 2 0

To represent a multirate hold H mathematically, we decompose the input vector u into vectors with the fastest and slower hold rates, as follows:

˜u f

˜ 共k兲 D ˜ 共k兲 C y y2

Ã共k兲 ª A BuCh共k兲 0 Ah共k兲

Fig. 2 A periodic time-varying system consisting of a timeinvariant generalized plant, a multirate sampler S, and a multirate hold H

˜ª H:u

˜B 共k兲 u

for k = 0 , 1 , 2 , . . ., where the matrices in Eq. 共15兲 are obtained by straightforward calculation, as follows:

K

u¬

˜B 共k兲 2

˜ 共k兲 D ˜ 共k兲 D ˜ 共k兲 = C 2 22 2u

˜y 共k兲

~u

Ã共k兲

冤

ZA ZB2 ZBu

= C2 D22 D2u Cy Dy2 0

冥冋册 w2 u

共17兲

Here, matrices with bold capital letters are block-diagonal,2 consisting of the matrices in 共Eq. 共15兲兲; for example 共12兲 Aª

Here, the dimensions of us and u f are denoted by ns and n f , respectively, ⍀共k兲苸 Rns⫻ns is a diagonal matrix with diagonal entries of 0 or 1, and plays a similar role to ⌫共k兲 in Eq. 共8兲. If we feed the ith input signal of ˜us from the controller at time k, then 共i , i兲-entry of ⍀共k兲 is set to 1; otherwise, it is set to 0, resulting in the ith input at time k equal to the ith input at time k − 1. In the single-rate case, the hold degenerates to a static system u共k兲 = ˜u共k兲. We assume that the hold is periodic with a period Th, i.e.

冤

Ã共0兲 Ã共T − 1兲

冥

共18兲

and Z is a “shift” matrix, which is of compatible size with A and defined by

冤冥 0 ¯ 0 I

Zª

0

I

]

共19兲

I 0

⍀共k + Th兲 = ⍀共k兲,

k = 0,1,2, . . .

共13兲

Now, we suppose that the periods Ts and Th are rationally related, that is, their least common multiple T ª lcm共Ts,Th兲

共14兲

is an integer. Then, by combining Eqs. 共6兲, 共7兲, and 共11兲, we obtain a linear periodic time-varying system with the period T Journal of Dynamic Systems, Measurement, and Control

The vectors z2, w2, y, and u are considered as “lifted” signals of z2, w2, ˜y , and ˜u in 共Eq. 共15兲兲, respectively. By a combination of the theories in Refs. 关4,1,2兴, we can deduce the following equivalences: •

A time-invariant controller

2 Throughout this paper, we use bold capital letters to denote block-diagonal matrices.

MARCH 2010, Vol. 132 / 021005-3


冋

u=

ZKA ZKB KC

KD

册

z

共20兲

y

∆ V

z

stabilizes the time-invariant system 共17兲, and satisfies a H2 or H⬁ norm condition 储Tz2w2储i ⬍

再

冑T␥2 ,

if i = 2

␥,

if i = ⬁

冎

共21兲

where T is defined in Eq. 共14兲. Here, the matrices in Eq. 共20兲 are block-diagonal, and we denote them as KM ª

冤

K M 共0兲 K M 共T − 1兲

冥

冋

˜u共k兲

册冋 =

KA共k兲 KB共k兲 KC共k兲 KD共k兲

册冋册 xK共k兲 ˜y 共k兲

i = 2 or ᐉ2-induced

共24兲

Robust Controller Design

In this section, we will present three robust controller design methods useful for robust track-following: mixed H2 / H⬁, mixed H2 / ␮, and robust H2 syntheses. These methods are based on convex optimization involving LMIs, to which there are numerically efficient algorithms 关5兴 and software 关6–8兴 available. Some of the LMIs, which are necessary to solve the optimization problems, will be given in the Appendixes A and B. Advantages and disadvantages of each method will be summarized. 4.1 Mixed H2 Õ Hⴥ Synthesis. The mixed H2 / H⬁ synthesis is a well-known design method for reconciling performance and robustness 关9,10兴. Since this approach can deal with only unstructured dynamic uncertainties, we will ignore parametric uncertainties in ⌬. In this approach, we can guarantee only robust stability for individual, not simultaneous, perturbations of ⌬V and ⌬ M 共see Fig. 3兲. Denote the set of all controllers that stabilize the closed-loop system in Fig. 3 for ⌬V 苸 BH⬁ and for ⌬ M 苸 BH⬁ by KV and K M , respectively. Then, the control problem in this approach is formally stated: as follows: PROBLEM 2. For given multirate sampler S and hold H with fixed sampling and hold rates, design a controller K that exponentially stabilizes the closed-loop system for all ⌬V 苸 BH⬁ and ⌬ M 苸 BH⬁, and minimizes the nominal rms value of z2 against Gaussian white noise w2, or equivalently, solve the optimization problem min

储Tz2w2共HKS,0兲储2

021005-4 / Vol. 132, MARCH 2010

y

M

w


2

2

u

K

H

This problem can be rewritten, as follows:

冦

共23兲

1. Derive a time-invariant system 共17兲. 2. Design a controller 共20兲 for Eq. 共17兲. 3. Obtain a periodic time-varying controller 共23兲 by decomposing controller matrices as Eq. 共22兲.

K苸KV艚K M

w

Fig. 3 Uncertainty structure for mixed H2 / Hⴥ synthesis

In this way, the ᐉ2 seminorm or ᐉ2-induced norm suboptimal control problem for a periodic time-varying system can be transformed into a standard H2 or H⬁ suboptimal control problem for a time-invariant system, with a controller structure 共Eqs. 共20兲 and 共22兲兲. The controller structure can be guaranteed by solving the suboptimal control problems with numerical tools for LMIs. To summarize, our procedure to solve the multirate control problem is

4

M

V

共22兲

of the period T stabilizes exponentially the time-varying system 共15兲, and satisfies a norm condition 储Tz2w2储i ⬍ ␥,

∆ M

S

where M can be A, B, C, or D, and the block sizes in Z are compatible with the block sizes in KA. • A periodic time-varying controller xK共k + 1兲

w V

共25兲

储Tz2w2共HKS,0兲储2 ⬍ ␥

min ␥, subject to 储TzVwV共HKS兲储ᐉ2 ⬍ 1 K

储TzM wM 共HKS兲储ᐉ2 ⬍ 1

冧

共26兲

where zV, z M , wV, and w M are signals shown in Fig. 3, and 储 · 储ᐉ2 means the ᐉ2-induced norm. We can solve the optimization 共26兲 by following the procedure given at the end of Sec. 3. An auxiliary time-invariant system corresponding to Eq. 共17兲 can be expressed in this case as

G:

冤冥冤冥 wV

zV

wM

zM

w2

→

u

冤

z2 y

ZA ZBV ZB M ZB2 ZBu CV DVV DVM DV2 DVu

G共z兲 ª C M D MV D MM D M2 D Mu C2 D2V D2M D22 D2u Cy

DyV

DyM

Dy2

0

冥

共27兲

where all the block matrices are obtained by the transformation process from multirate control to time-invariant control presented in Sec. 3. Using the system matrices, the inequality conditions in Eq. 共26兲 can be expressed as LMI conditions 关11,12兴, which are given in Appendix B.1. Main advantages of the mixed H2 / H⬁ method in trackfollowing control are that the computational cost involved in finding a solution to Eq. 共26兲 is relatively low, and that an optimal solution is guaranteed to be obtained given a feasible condition. While the robust H2 synthesis methodology, which will be presented later, involves the iterative solution of convex optimization problems in order to solve for a nonconvex one, the mixed H2 / H⬁ method requires the solution of just one convex problem. However, a disadvantage of this method is that it can neither deal with structured uncertainties, nor guarantee robust performance. In other words, there is no guarantee that robustness in our original problem 共Problem 1兲 is indeed satisfied by the mixed H2 / H⬁ design. 4.2 Mixed H2 Õ ␮ Synthesis. The ␮-theory is a powerful tool to deal with various types of uncertainties in robust control 关13兴. If the performance is evaluated with the ᐉ2-induced 共or H⬁兲 norm, one can design a controller for robust performance via the socalled D-K iterations in the ␮-synthesis theory. However, since the performance measure in the present problem is the ᐉ2 seminorm 共or H2 norm兲, we cannot use the ␮-synthesis theory as it is. Here, we combine the D-K iteration with the mixed H2 / H⬁ synTransactions of the ASME


thesis to design a controller to achieve nominal performance optimization with guaranteed robust stability. To be more precise, we consider the following problem: PROBLEM 3. For given multirate sampler S and hold H with fixed sampling and hold rates, design a controller K that stabilizes the closed-loop system for all ⌬ 苸 B, and minimizes the nominal rms value of z2 against Gaussian white noise w2, i.e., solve the optimization problem min 储Tz2w2共HKS,0兲储2

共28兲

K苸K共B兲

Notice that the only difference between Eqs. 共28兲 and 共5兲 is that the term max⌬苸B is not taken into consideration in Eq. 共28兲, and thus we optimize only nominal performance. ˜ is a single-rate Let us first consider a simpler situation, where K ˜ controller. Then, by ␮-analysis theory 关13,14兴, the condition K ˜ 苸 K共B兲 can be guaranteed by K 苸 K共0兲共nominally stabilizing兲 and ˜ 兲D−1储 ⬍ 1 储DTz⌬w⌬共K ⬁

共29兲

for some matrix function D 苸 D. Here, D is a set of matrix functions associated with B in Eq. 共4兲 and defined by Dª

再

D ª diag关d1, . . . ,d p+2兴 d j 苸 H⬁,d−1 j 苸 H⬁, j = 1, . . . ,p + 2

冎

共30兲

Therefore, problem 共28兲 can be solved via mixed H2 / H⬁ optimization with D-scaling min ˜ 苸K共0兲,D苸D K

再

␥, subject to

˜ ,0兲储 ⬍ ␥ 储Tz2w2共K 2 ˜ 兲D−1储 ⬍ 1 储DTz⌬w⌬共K ⬁

冎

1. Design a rational matrix D 苸 D, as well as a single-rate con˜ with the fastest sampling/hold rate 共or a troller K continuous-time controller if a generalized plant is given in continuous-time兲, that solves the optimization inf

˜ 兲D−1储储DTz⌬w⌬共K ⬁

4.3 Robust H2 Synthesis. The previous two methods cannot handle robust performance, and therefore, plant perturbations may degrade the track-following performance to an unacceptable extent. The third method, which is based on the result in Ref. 关18兴, can guarantee robust performance for parametric uncertainties, but not for dynamic ones. Define a set B p of parametric uncertainties as B p ª 兵⌬ ª diag关␦1, . . . , ␦ p兴, ␦ j 苸 BR, j = 1, . . . ,p其.

共32兲

K苸K共0兲

再

储Tz2w2共HKS,0兲储2 ⬍ ␥ 储DTz⌬w⌬共HKS兲D−1储ᐉ2 ⬍ 1

冎

共33兲

We remark that the constraint 储DTz⌬w⌬共HKS兲D 储ᐉ2 ⬍ 1 −1

max 储Tz2w2共HKS,⌬兲储2

共34兲

guarantees closed-loop stability for all ⌬ 苸 B, due to the Small Gain Theorem. The optimization problem 共32兲 in step 1 can be solved via D-K iteration with “␮-Analysis and Synthesis Toolbox” 共or the latest “Robust Control Toolbox”兲 in MATLAB 关15兴. On the other hand, since the optimization problem 共33兲 is of the same type as 共26兲, the technique outlined in Sec. 4.1 can be used to solve the optimization in step 2. The main advantage of the proposed mixed H2 / ␮ synthesis, as compared with the mixed H2 / H⬁ synthesis, is the freedom in the selection of D. Notice that D = I corresponds to the mixed H2 / H⬁ design. Because of this freedom, we can expect better nominal performance and robust stability. However, one disadvantage is that the problem becomes nonconvex, and therefore, the obtained solution may not be a global optimum. Moreover, the solution Journal of Dynamic Systems, Measurement, and Control

共36兲

K苸K共B p兲 ⌬苸B p

To use the result in Ref. 关18兴, we need that the following assumptions are satisfied. ASSUMPTION 1. D⌬⌬ = 0 and Dy⌬ = 0 in Eq. (1). The assumption D⌬⌬ = 0 guarantees that the closed-loop system matrices depend on ⌬ affinely 共see Eq. 共38兲兲, while Dy⌬ = 0 ensures the well-posedness of the closed-loop system by forcing the direct term from u to y to be zero 共see Eq. 共37兲兲. These assumptions are not restrictive, since they hold most of the trackfollowing problems in HDDs. In the case with only parametric uncertainties, the uncertain system from 关wT2 , uT兴T to 关zT2 , y T兴T is obtained as

冋册

2. Using the D obtained in Step 1, solve a multirate mixed H2 / H⬁ problem min ␥, sub. to

共35兲

The design problem in this section is stated next. PROBLEM 4. For given multirate sampler S and hold H with fixed sampling and hold rates, design a controller K that stabilizes the closed-loop system for all ⌬ 苸 B p, and minimizes the worstcase rms value of z2 against Gaussian white noise w2 for all ⌬ 苸 B p, i.e., solve min

共31兲

Using the single-rate result, it is possible to solve the above multirate control problem approximately, with the following procedures:

˜ 苸K共0兲 D苸D,K

may be computationally expensive to obtain, especially when we select a D of high order. In addition, like the mixed H2 / H⬁ approach, there is no guarantee regarding robust performance. Remark 2. Since our uncertainty set B includes parametric uncertainties, we can have a less conservative robust stability condition than Eq. 共29兲共see Refs. 关14,16,17兴兲. If we use the less conservative condition, an extra “G-scaling” must be introduced. This will increase the computational effort, as well as the final controller order. Remark 3. It has been our experience that the mixed H2 / ␮ controller performs better if the performance channel is included in solution to step 1, even though the measure in ␮-synthesis is not the H2 norm but rather the H⬁ norm.

z2 y

=

冤

A⌬ B⌬2

B⌬u

C⌬2

D⌬2u

D⌬22

0

Cy Dy2

冥冋册 w2

共37兲

u

where the superscript ⌬ means a “matrix with uncertainties,” and the system matrices are given by

冋

A⌬ B⌬2

B⌬u

C⌬2

D⌬2u

D⌬22

册冋 ª

+

A

B2

Bu

C2 D22 D2u

冋册关 B⌬

D2⌬

册

⌬ C⌬ D⌬2 D⌬u 兴

共38兲

In the robust H2 method that will be explained below, it is important that the system matrices in Eq. 共38兲 are affine in ⌬, and that ⌬ belongs to a convex polyhedron B p 共which is a hypercube in the present setting兲. To deal with the multirate characteristics of the controller, we use the procedure in Sec. 3. Then, for an augmented timeinvariant plant

冋册 z2 y

=

冤

ZA⌬ ZB⌬2 ZB⌬u C⌬2

D⌬22

D⌬2u

Cy

Dy2

0

冥冋

w2 u

册

共39兲

we need to design a linear time-invariant controller of the form MARCH 2010, Vol. 132 / 021005-5


u=

冋

ZKA ZKB KC

KD

册

10

共40兲

y

ª

冋

Ccl共⌰,⌬兲 Dcl共⌰,⌬兲

Z共A⌬0 + B⌬⌰C兲 C⌬0 + D⌬12⌰C

Z共B⌬0 + B⌬⌰D21兲 D⌬22 + D⌬12⌰D21

10

册册

(F)

0

torsion1

2

10 10

10

3

冋册 0 Z

⌰ª

,

⌬

0 B⌬u

B ª

Cª

0

I

0

I

Cy 0

KA KB

10

,

0

0

,

B⌬0

D21 ª

10

Dy2

⌬苸B p

This can be solved by the following optimization, which involves a finite number of matrix inequalities:

␥ ⬎ traceW W Ccl共⌰,⌬k兲 Dcl共⌰,⌬k兲 ⴱ 0 P ⬎0 ⴱ ⴱ I min ␥, subject to W,P,⌰ PZ PZAcl共⌰,⌬k兲 PZBcl共⌰,⌬k兲 ⴱ 0 P ⬎0 ⴱ ⴱ I

冥

冥

冧

共41兲

for ⌬k 苸 V共B p兲. Here, the matrices P and W are block-diagonals of appropriate sizes, PZ ª ZTPZ, V共B p兲 is the set of all vertices of a convex polyhedron B p, and the “ⴱ-blocks” are the block matrices that make the total matrix symmetric. The replacement of infinitely many inequality constraints for ⌬ 苸 B p with finitely many ones at vertices ⌬k 苸 B p is possible due to the facts that the closedloop system matrices are affine in ⌬, and that the set B p is a convex polyhedron. Unfortunately, this problem is nonconvex, since there are coupling terms between P and ⌰ in Eq. 共41兲. However, by using the coordinate descent method 共see Refs. 关19,20兴 and references therein兲, we can find a local optimum. The procedure is presented next. [Initial design of ⌰兴 This will be explained below, as well as in Appendix B.2. Set the result of the initial design to ⌰1. Also, set i = 1. [Design of P兴 Fix ⌰ ª ⌰i. Solve the convex optimization problem 共41兲 with respect to ␥, W, and P. Set a solution P to Pi. [Design of ⌰兴 Fix P ª Pi. Solve the convex optimization problem 共41兲 with respect to ␥, W, and ⌰. Set a solution ⌰ to ⌰i+1. Increment i by one. Continue this iteration until convergence. 021005-6 / Vol. 132, MARCH 2010

10

3

frequency [Hz]

4

10

0

min max 储Tcl共⌰,⌬兲储2

冦

2

Fig. 4 Frequency responses from uV to yLDV „upper figure… and from uM to yLDV „lower figure…

0

Our problem is to find a robustly stabilizing controller matrix ⌰ that solves

冤冤

(T2)

0

10

C⌬0 ª 关C⌬2 0 兴, D⌬12 ª 关0 D⌬2u 兴

⌰

2

torsion1 (T1) torsion2

B⌬2

ª

4

10

sway

KC KD

A⌬ 0

A⌬0 ª

(T2)

(S)

冋册冋册冋册冋册冋册冋册 Z 0

torsion2

4

where the matrices are defined by Zª

(S)

(B)

(T1) magnitude

冋

Acl共⌰,⌬兲 Bcl共⌰,⌬兲

sway

butterfly friction

We remark that all the uncertain matrices in Eq. 共39兲 are affine in ⌬. The time-invariant closed-loop system of Eqs. 共39兲 and 共40兲 is expressed as 共Tcl共⌰,⌬兲兲共z兲 ª

5

Remark 4. Theoretically, since the value ␥ is monotonically nonincreasing during the iterations and it has a lower bound 共which is 0兲, it will converge to some positive number. Numerically, we stop the iteration when the value of ␥ stops decreasing 共within some tolerance兲. Generally speaking, in nonconvex optimization problems, the selection of an initial point is critical. We follow the procedure given in Ref. 关18兴 to derive a reasonable initial point, which will be reviewed in Appendix B.2. The advantage of the robust H2 synthesis over the other synthesis methodologies discussed in this paper is the ability to cope with robust performance in the design. However, to utilize the proposed synthesis technique, it is necessary to ignore dynamic uncertainties, which generally capture high-frequency unmodeled dynamics, typically unavoidable in modeling. In addition, the computation of a solution to the robust H2 problem is demanding, because we have to solve for a series of convex optimization problems iteratively in order to solve the nonconvex problem. Further, the number of inequality constraints increases exponentially with the number of parametric uncertainties, since the constraints are imposed at vertices of a hypercube B p. Therefore, only a few parametric uncertainties can be included in a practical design.

5

A Design Example

In this section, we will demonstrate the usefulness of the robust H2 synthesis method presented in Sec. 4.3 through one simple example of a dual-stage track-following control problem. This example is taken from Ref. 关21兴. Other examples are shown in Ref. 关3兴. The system to be considered is a PZT-actuated dual-stage servo system, whose inputs are the VCM input 共uV兲 and the input to the PZT-actuators 共u M 兲, and whose output is the head position 共y LDV兲 measured by laser Doppler vibrometer 共LDV兲. Readers are referred to Refs. 关21–23兴 and their references for details on the identification, modeling, and control of the PZT-actuated dualstage servo system discussed in this section. 5.1 Continuous-Time Modeling. In order to use the robust H2 synthesis technique, we need a mathematical model with only parametric uncertainties. To derive such a mathematical model, 36 frequency responses of PVCM 共from uV to y LDV兲 and of PMA 共from u M to y LDV兲 have been taken in Ref. 关21兴, as shown in Fig. 4. To reduce the computational burden during the controller synTransactions of the ASME


V

u

P M

P

E

+

P +

D

y

m a

δ 1

δ 2Ι

L D V

δ 3Ι

Fig. 5 Block diagram V

thesis, it is advantageous to build a model with its order and the number of uncertain parameters as small as possible. As shown in Fig. 4, since the suspension’s sway mode 共S兲 and torsion modes 共T1 and T2兲 can be seen in both frequency responses, these suspension modes can be factored out as a common transfer function Pma in the block diagram in Fig. 5 PVCM共s兲 = Pma共s兲PE共s兲, PMA共s兲 = Pma共s兲PD共s兲

共43兲

Pma共s兲 = PT1共s兲PS共s兲PT2共s兲

共44兲

where each term of the right-hand sides is of the form b0j s2 + b1j s + b2j s2 + a1j s + a2j

,

j = F,B,T1,S,T2

Pc =

再

P共s兲 = D + C共sI − A共⌬兲兲−1B共⌬兲 ⌬ = diag关␦1, ␦2I, ␦3I兴, 兩␦i兩 ⱕ 1, ∀ i

冎

共50兲

Since there are three uncertain parameters ␦i, the set Pc has 23 “extreme” cases denoted by ⌬i, where each parameter ␦i takes the value of ⫺1 or 1. For each extreme case, we transform the continuous-time model into a discrete-one by zero-order hold, yielding discrete-time 共A , B兲-matrices as follows:

共45兲

Aid ª eA共⌬i兲T,

with uncertain coefficients

Bid ª

冕

T

eA共⌬i兲␶d␶B共⌬i兲

0

j j ␦ 兲, bkj = ¯bkj共1 + M bk

k = 1,2

共46兲

j j ␦ 兲, akj = ¯akj共1 + M ak

k = 1,2

共47兲

j Here, the nominal values are denoted by ¯akj and ¯bkj, and M ak and j M bk are the constant weightings. It has been found that the experimental frequency responses can be reproduced accurately enough by using only three uncertain parameters

␦1 = ␦ , ␦2 = ␦ , ␦3 = ␦ = ␦ = ␦ F

L D V

Fig. 6 Block diagram with parametric uncertainties

共42兲

PE共s兲 = PF共s兲PB共s兲

y

n o m

M

Here, the E-block PE and suspension Pma dynamics are assumed to have structures as

P j共s兲 =

P

B

T1

T2

S

Pd ª 兵P共z兲 = D + C共zI − A兲−1B, 关A,B兴苸 B其 where Bª

共48兲

PD is the piezoelectric actuator driver dynamics that is not observed in PVCM, and of the form D PD共s兲 = bD 1 s + b2

where T = 25⫻ 10−6 共s兲 is a sampling period. For controller design, we use all the convex combinations of these eight discretetime 共A , B兲-matrices:

共49兲

Assuming the structure of the model 共42兲–共49兲, the nominal values and the weighting coefficients are identified as in Table 1. The validity of the modeling result will be examined after the discretization of the model set in Sec. 5.2. 5.2 Discretization of a Continuous-Time Model Set. For digital controller design, the continuous-time uncertain model is transformed into a discrete-time one. Although there are several ways to carry out such transformation, we present only one method here. The block diagram in Fig. 5 can be expressed as a linear fractional transformation 共LFT兲 form with uncertain parameters, as in Fig. 6. We can represent the continuous-time model set from 关uV , u M 兴T to y LDV as

再

8

关A,B兴 =

兺 ␣ 关A ,B 兴, k

i d

i d

8

␣i ⱖ 0,

i=1

兺␣ =1 i

i=1

冎

The comparisons between experimental frequency responses and those of sampled models in the model set Pd are shown in Figs. 7 and 8. Both figures show that the observed dynamic variation can be accurately represented by the parametric uncertainty modeling technique. 5.3 Controller Design via Robust H2 Synthesis. To determine a controller design problem, we need to specify the variables z2, w2, y, and u in Eq. 共37兲. In this example, control inputs u are taken as u ª 关uV,u M 兴T As a disturbance, in this simple example, we consider only track runout r, modeled as r = W rw 2 where Wr is a shaping filter

Table 1 Nominal parameters and weighting coefficient values Mode j

¯b j 0

¯b j 1

¯b j 2

¯a1j

¯a2j

j M b1

j M b2

j M a1

j M a2

F B T1 S T2 D

0 0 1 0 1 –

0 0 720 0 6300 4.0026⫻ 10−4

4.206⫻ 108 1.003⫻ 109 1.598⫻ 109 3.316⫻ 109 1.079⫻ 1010 47.3151

51.175 569.8 208.2 1015 2700 –

2.365⫻ 104 1.003⫻ 109 1.575⫻ 109 3.316⫻ 109 1.02⫻ 1010 –

0 0 ⫺0.03 0 ⫺0.01

0.3 0.35 ⫺0.03 0.12 ⫺0.01

0 0 0 0.1 0

0 0.35 0.05 0.12 0.05

Journal of Dynamic Systems, Measurement, and Control

MARCH 2010, Vol. 132 / 021005-7


2

Experimental frequency responses

10

0

0

10

magnitude

magnitude

10

−2

10

4

10 2

Model set frequency responses

10

−1

10

−2

magnitude

10 0

10

−3

10

2

3

10

−2

10

10 frequency [Hz]

4

10

4

10

frequency [Hz]

Fig. 9 Sensitivity functions Fig. 7 Comparisons between the experimental and simulation frequency responses for PVCM

Wr共s兲 =

3.162s + 1.987 ⫻ 105 s + 628.3

and w2 is white noise 共this w2 corresponds to the one in Eq. 共37兲兲. The difference between r and y LDV e ª r − y LDV is the PES, which is the measurement y in Eq. 共37兲. As the control outputs z, we use the vector consisting of the PES and weighted input signals as z ª 关e,QVuV,Q M u M 兴

T

where the weights QV and Q M are selected by trial-and-error as QV = 0.1 and Q M = 5 ⫻ 10−5. The designed controller was of order 13. In order to analyze the controller, the sensitivity functions for 100 sampled models are shown in Fig. 9. As can be seen in the figure, the sensitivity functions do not disperse even in the face of parameter variations, indicating that the obtained controller indeed satisfies its intended robust performance property. This property was also verified during the experimental tests, as discussed in Ref. 关21兴. The shaping of the closed-loop sensitivity transfer function can

4

Experimental frequency responses

magnitude

10

2

10

0

10

4

10 4

Model set frequency responses

magnitude

10

2

6

Conclusions

In this paper, we have presented three multirate robust controller design methods for track-following control in dual-stage multisensing servo systems. The procedures, advantages, and disadvantages have been provided for each method. In addition, multirate control problems have been shown to be reduced to time-invariant control problems. All of the methods rely on numerically efficient solvers for LMIs. This paper can be seen as a collection of results from the robust control literature, including those in Refs. 关1,2,18,13兴, which are specially useful for trackfollowing controller design in dual-stage HDDs. One trackfollowing control example for a PZT-actuated suspension dualstage system was given to illustrate that the robust H2 controller is useful in maintaining the track-following performance under plant perturbations. It is known that one major drawback of robust control theory is that the designed controllers typically have high orders. Therefore, controller reduction should also be done after all of the three proposed design procedures. Since a multirate controller is periodic time-varying in general, we can utilize the existing balanced truncation techniques, e.g., those in Refs. 关26,27兴, for the reduction purpose. Some realistic examples in Ref. 关3兴 show that our design methods plus model reduction are quite promising in designing low order controllers with robust track-following performance. Finally, fixed-order controller design such as the one presented in Ref. 关20兴 will be useful in this application, which is a future research topic.

Acknowledgment

10

0

10

be achieved by properly selecting the runout model. The selection of the runout model is in fact a design parameter in our synthesis methodologies, and the one selected lead to a reasonable sensitivity function. In general, the magnitude frequency response of the sensitivity function tends to follow the inverse of the magnitude of the runout model, subjected to the constraints imposed by the other weighting terms in the H2 cost function and the robustness constraints 关22兴. The systematic and optimal design of a runout model from actual PES data is an important research subject, which is actively carried out by the disk drive companies 关24,25兴.

4

10 frequency [Hz]

Fig. 8 Comparisons between the experimental and simulation frequency responses for PMA

021005-8 / Vol. 132, MARCH 2010

The authors are grateful to Professor Raymond de Callafon at the Mechanical Engineering Department of the University of California, San Diego, CA, for providing the experimental data used in the dual-stage servo example presented in this paper. This work was supported by the Information Storage Industry Consortium 共INSIC兲, the Computer Mechanics Laboratory 共CML兲 at UC Berkeley, the Swedish Research Council 共VR兲 and the National SciTransactions of the ASME


ence Foundation under Grant No. CMS-0428917.

appropriate block sizes, we define

Here, we present LMIs used for our mixed H2 / H⬁ design in Sec. 4.1. These LMIs can be derived by combining the ideas of congruent transformations in Ref. 关11兴 and of synthesis for timevarying systems in Refs. 关1,4兴. We use the following notation: For a block-diagonal matrix M and a shift matrix Z 共see Eq. 共19兲兲 with

min traceW,subject to

冤

冤 XZ

I

XZ

I

冤

Note that MZ is also block-diagonal. Using the system matrices in Eq. 共27兲 and block diagonal deciˆ ,K ˆ ,K ˆ , and K ˆ with sion variables W = WT, X = XT, Y = YT, K A B C D appropriate sizes, the optimization problem Eq. 共26兲 is equivalent to

W C2 + D2uKˆ DCy C2Y + D2uKˆ C D22 + D2uKˆ DDy2 ⴱ

X

I

0

ⴱ

ⴱ

Y

0

ⴱ

ⴱ

ⴱ

I

XZA + Kˆ BCy

XZB2 + Kˆ BDy2

Kˆ A

ⴱ

YZ A + BuKˆ DCy AY + BuKˆ C B2 + BuKˆ DDy2

ⴱ

ⴱ

X

I

0

ⴱ

ⴱ

ⴱ

Y

0

ⴱ

ⴱ

ⴱ

ⴱ

I

ˆ C X ZA + K B y

Kˆ A

冥

XZBi + Kˆ BDyi

0

Bi + BuKˆ DDyi

0

YZ A + BuKˆ DCy AY + BuKˆ C

ⴱ

ⴱ

X

I

0

T T Diu CiT + CTy Kˆ D

ⴱ

ⴱ

ⴱ

Y

0

ˆ T DT YCiT + K C iu

ⴱ

ⴱ

ⴱ

ⴱ

T T DiiT + DTyiKˆ D Diu

ⴱ

ⴱ

ⴱ

ⴱ

ⴱ

where i = V , M. Note that all the entries in the above LMIs are block-diagonal. For the controller reconstruction, we first compute for block-diagonal nonsingular matrices M and N, having the same block structure as X and Y, and satisfying 共A3兲

I

共A4兲

ˆ C Y兲N−T KC = 共Kˆ C − K D y

共A5兲

KB =

MZ−1共Kˆ B

− XZB2Kˆ D兲

共A6兲

As has been explained, the robust H2 controller is designed via nonconvex optimization 共41兲. For nonconvex optimization, selecJournal of Dynamic Systems, Measurement, and Control

共A2兲

共B1兲

u = K cx that optimizes robust H2 performance max 储Tz2w2储2

共B2兲

⌬苸B p

As given in theorem 6 in Ref. 关18兴, this problem can be solved by convex optimization with LMIs:

共A7兲

Appendix B: Initial Controller Design in Robust H2 Synthesis

冥

⬎0

B.1 State Feedback. First, for the uncertain system 共39兲, we design a state feedback controller as

ˆ − X 共A + B Kˆ C 兲Y − X B K NT − M K C Y其N−T KA = MZ−1兵K A Z u D y Z u C Z B y Note that the block structures of KA, KB, KC, and KD are guaranteed in the computations, since all the matrices in the right-hand sides of Eqs. 共A4兲–共A7兲 are block-diagonal. The corresponding periodic time-varying controller can be obtained by dividing these block matrices, as in Eq. 共22兲.

⬎0

tion of an initial point is of great importance. Here, we review a reasonable method for such selection proposed in Ref. 关18兴.

The controller matrices are given by ˆ KD = K D

冥

⬎0

ⴱ

MNT = I − XY

共A1兲

MZ ª ZTMZ

Appendix A: LMIs for Mixed H2 Õ Hⴥ Synthesis

冦

min traceW, subject to W,Q,L

冤冤

冥冥

W C⌬2 Q + D⌬2uL D⌬22 ⴱ

Q

0

ⴱ

ⴱ

I

QZ A⌬Q + B⌬u L B⌬2 ⴱ

Q

0

ⴱ

ⴱ

I

⬎0

⬎0

冧

共B3兲

where the constraints are imposed at the vertices of B p, ⌬ = ⌬k 苸 V共B p兲. Using the solutions L and Q, the optimal state feedback is given by KC ª LQ−1

共B4兲

MARCH 2010, Vol. 132 / 021005-9


B.2 Output Feedback. Next, by fixing KC in Eq. 共B4兲 and KD = 0, we design an output feedback u=

冋

ZKA ZKB KC

0

册

共B5兲

y

冦冤冤

min

traceW, subject to

W,X,Y,U,V

that optimizes robust H2 performance 共B2兲. Define AF⌬ ª A⌬ + B⌬u KC

Then, due to Theorem 8 in Ref. 关18兴, this problem can be solved by convex optimization with LMIs:

⌬ ⌬ ⌬ W Cz⌬ + Dzu KC Dzu KC Dzw

ⴱ

X

0

0

ⴱ

ⴱ

Y

0

ⴱ

ⴱ

ⴱ

I

XZAF⌬

− XZB⌬u KC

XZBw⌬

ⴱ

YZ YZAF⌬ − V − UCy V − YZB⌬u KC YZBw⌬ − UDyw

ⴱ

ⴱ

X

0

0

ⴱ

ⴱ

ⴱ

Y

0

ⴱ

ⴱ

ⴱ

ⴱ

I

关13兴关14兴

References 关1兴 Dullerud, G. E., and Lall, S., 1999, “A New Approach for Analysis and Synthesis of Time-Varying Systems,” IEEE Trans. Autom. Control, 44共8兲, pp. 1486–1497. 关2兴 Lall, S., and Dullerud, G., 2001, “An LMI Solution to the Robust Synthesis Problem for Multi-Rate Sampled-Data Systems,” Automatica, 37共12兲, pp. 1909–1922. 关3兴 Huang, X., Nagamune, R., and Horowitz, R., 2006, “A Comparison of Multirate Robust Track-Following Control Synthesis Techniques for Dual-Stage and Multisensing Servo Systems in Hard Disk Drives,” IEEE Trans. Magn., 42共7兲, pp. 1896–904. 关4兴 Peters, M. A., and Iglesias, P. A., 1997, Minimum Entropy Control for TimeVarying Systems. Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA. 关5兴 Nesterov, Y., and Nemirovskii, A., 1994, Interior-Point polynomial Algorithms in Convex Programming, SIAM, Philadelphia, PA. 关6兴 Sturm, J. F., 1999, “Using SEDUMI 1.02, a MATLAB Toolbox for Optimization Over Symmetric Cones,” 共Special Issue on Interior Point Methods兲, Optim. Methods Software, 11, pp. 625–653. 关7兴 Gahinet, P., Nemirovski, A., Laub, A. J., and Chilali, M., 1995, LMI Control Toolbox User’s Guide, The MathWorks, Inc., Natick, MA. 关8兴 Löfberg, J., 2004, “YALMIP: A Toolbox for Modeling and Optimization in MATLAB,” Proceedings of the CACSD Conference, Taipei, Taiwan. Available from http://control.ee.ethz.ch/~joloef/yalmip.php 关9兴 Bernstein, D. S., and Haddad, W. H., 1989, “LQG Control With an H⬁ Performance Bound: A Riccati Equation Approach,” IEEE Trans. Autom. Control, 34共3兲, pp. 293–305. 关10兴 Khargonekar, P., and Rotea, M., 1991, “Mixed H2 / H⬁ Control: A Convex Optimization Approach,” IEEE Trans. Autom. Control, 36, pp. 824–837. 关11兴 Scherer, C., Gahinet, P., and Chilali, M., 1997, “Multiobjective OutputFeedback Control via LMI Optimization,” IEEE Trans. Autom. Control, 42共7兲, pp. 896–911. 关12兴 Masubuchi, I., Ohara, A., and Suda, N., 1998, “LMI-Based Controller Synthe-

021005-10 / Vol. 132, MARCH 2010

⬎0

0

共B8兲

KB ª YZ−1U

冥

XZ

where the constraints are again imposed at the vertices of B p, ⌬ = ⌬k 苸 V共B p兲. Using the optimizers, KA and KB are obtained as KA ª YZ−1V,

共B6兲

关15兴关16兴关17兴关18兴关19兴关20兴关21兴关22兴关23兴

关24兴关25兴关26兴关27兴

冥

共B7兲

sis: A Unified Formulation and Solution,” Int. J. Robust Nonlinear Control, 8共8兲, pp. 669–686. Packard, A., and Doyle, J. C., 1993, “The Complex Structured Singular Value,” Automatica, 29共1兲, pp. 71–109. Young, P. M., 1996, “Controller Design With Real Parametric Uncertainty,” Int. J. Control, 65共3兲, pp. 469–509. Balas, G., Chiang, R., Packard, A., and Safonov, M., 2005, Robust Control Toolbox for Use With MATLAB, The Mathworks, Inc., Natick, MA. Young, P. M., and Doyle, J. C., 1996, “Properties of the Mixed ␮ Problem and Its Bounds,” IEEE Trans. Autom. Control, 41共1兲, pp. 155–159. Tits, A. L., and Chou, Y. S., 2000, “On Mixed-␮ Synthesis,” Automatica, 36, pp. 1077–1079. Kanev, S., Scherer, C., Verhaegen, M., and De Schutter, B., 2004, “Robust Output-Feedback Controller Design via Local BMI Optimization,” Automatica, 40, pp. 1115-1127. Iwasaki, T., and Skelton, R. E., 1995, “The XY-Centering Algorithm for the Dual LMI Problem: A New Approach to Fixed Order Control Design,” Int. J. Control, 62共6兲, pp. 1257–1272. Iwasaki, T., 1999, “The Dual Iteration for Fixed-Order Control,” IEEE Trans. Autom. Control, 44共4兲, pp. 783–788. de Callafon, R. A., Nagamune, R., and Horowitz, R., 2006, “Robust Dynamic Modeling and Control of Dual-Stage Actuators,” IEEE Trans. Magn., 42共2兲, pp. 247–54. Li, Y., and Horowitz, R., 2002, “Design and Testing of Track-Following Controllers for Dual-Stage Servo Systems With PZT Actuated Suspensions,” Microsyst. Technol., 8, pp. 194–205. Li, Y., Marcassa, F., Horowitz, R., Oboe, R., and Evans, R., 2003, “TrackFollowing Control With Active Vibration Damping of a PZT-Actuated Suspension Dual-Stage Servo System,” Proceedings of the American Control Conference, Vol. 3. Abramovitch, D., Hurst, T., and Henze, D., 1998, “An Overview of the PES Pareto Method for Decomposing Baseline Noise Sources in Hard Disk Position Error Signals,” IEEE Trans. Magn., 34共1兲, pp. 17–23. Ehrlich, R., and Curran, D., 1999, “Major HDD TMR Sources and Projected Scaling With TPI,” IEEE Trans. Magn., 35, pp. 885–891. Varga, A., 2000, “Balanced Truncation Model Reduction of Periodic Systems,” Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, pp. 2379–2384. Sandberg, H., and Rantzer, A., 2004, “Balanced Truncation of Linear TimeVarying Systems,” IEEE Trans. Autom. Control, 49共2兲, pp. 217–229.

Transactions of the ASME


Robust Control Synthesis Techniques for Multirate and Multisensing ...

Robust Control Synthesis Techniques for Multirate and Multisensing ...

Suggest Documents