Iterative Data-Driven H∞ Norm Estimation of Multivariable Systemswith Application to Robust Active Vibration Isolation

Tom Oomen, Rick van der Maas, Cristian R. Rojas, and Hakan Hjalmarsson

Abstract—This paper aims to develop a new data-drivenH∞ norm estimation algorithm for model-error modeling ofmultivariable systems. An iterative approach is presented thatrequires significantly fewer prior assumptions on the true system,hence it provides stronger guarantees in a robust control design.The iterative estimation algorithm is embedded in a robustcontrol design framework with a judiciously selected uncertaintystructure to facilitate high control performance. The approachis experimentally implemented on an industrial active vibrationisolation system (AVIS).

I. INTRODUCTION

Robustness is of key importance in any feedback controlledsystem. An important example of such a feedback controlledsystem is an active vibration isolation system (AVIS) [1],[2], which is used to isolate highly accurate motion systemsfrom external disturbances in multiple degrees-of-freedom.The underlying idea of vibration isolation is based on theconcept of skyhook damping, see [3]. Model-based controldesigns based on H∞-optimization are considered in [4]and [5]. In [4], model uncertainty is explicitly taken intoaccount. However, the uncertainty is based on inaccurate priorassumptions, leading to potentially conservative results.

Robustness analysis and robust control design hinge on anaccurate quantification of model quality. On the one hand,several model-error modeling techniques have been developedthat are based on existing system identification methods [6].For instance, in [7], the model error is quantified through theevaluation of the bias and variance errors of an estimated para-metric model of the model residual. Such an approach oftenrequires user intervention in the model parameterization stepand relies on assumptions that are typically asymptotic in thedata length. Alternatively, model-error modeling techniqueshave been proposed in [8], [9] that are based on non-parametricfrequency domain models. In [9], it is acknowledged that theuse of a discrete frequency grid, which is unavoidable inany finite time experiment, leads to intergrid errors. In [8],these errors are ignored by the assumption that the frequency

Tom Oomen and Rick van der Maas are with the Eindhoven Uni-versity of Technology, Department of Mechanical Engineering, Con-trol Systems Technology Group, PO Box 513, Building Gemini-Zuid,5600 MB Eindhoven, The Netherlands. Email: t.a.e.oomen@tue.nl,r.j.r.v.d.maas@tue.nl. Cristian Rojas and Hakan Hjalmarssonare with the Automatic Control Lab and ACCESS Linnaeus Cen-ter, Electrical Engineering, KTH - Royal Institute of Technology, S-100 44 Stockholm, Sweden. Email: cristian.rojas@ee.kth.se,hakan.hjalmarsson@ee.kth.se. This work was supported in part bythe European Research Council under the advanced grant LEARN, contract267381, in part by the Swedish Research Council under contracts 621-2009-4017 and 621-2011-5890, and in part by the Innovational Research IncentivesScheme under the VENI grant Precision Motion: Beyond the Nanometer(no. 13073) awarded by NWO (The Netherlands Organisation for ScientificResearch) and STW (Dutch Science Foundation).

grid is sufficiently dense. In [9], these intergrid errors arebounded by using prior assumptions, see also [10] for relatedapproaches. However, bounding these errors generally leadsto overly large estimates of the model error, as is also arguedin [11, Sec. 9.5.2]. Finally, model validation techniques havebeen proposed, see [12], [10] for frequency domain results and[13] for time domain results. However, such approaches oftenlead to overly optimistic estimates of the model errors unlessappropriate validation experiments are designed [14].

On the other hand, recently in [15, Sec. 12.2] a data-driven iterative procedure is proposed that direct estimatesthe H∞ norm of single-input single-output (SISO) systems.This enables its use in model-error modeling, since reliablerobust control design methodologies are available that considermodel errors as H∞ norm bounded perturbations. In [16], thedata-driven iterative H∞ norm estimation is further extended,followed by a thorough stochastic analysis in [17]. In [18], theprocedure is successfully applied for robust stability analysisof a SISO experimental setup.

Although the iterative data-driven H∞ norm estimationalgorithm has been thoroughly analyzed and successfully usedfor robustness stability analysis, it is not directly applicableto model-error modeling for achieving robust performance ofgeneral multivariable systems. First, the algorithms in [15,Sec. 12.2], [16], and [17] are only suitable for SISO systems.Second, besides the size of the uncertainty in terms of theH∞ norm, the uncertain model set highly depends on themodel uncertainty structure and frequency weighting func-tions. Indeed, typical model-error modeling procedures involvethe selection of a frequency dependent weighting function toshape the uncertainty, see [19, Sec. 7.4.3] for details in thisdirection. However, iterative data-driven H∞ norm estimationalgorithms do not provide the required frequency-dependentshape of the model error and cannot be used as a basis fordesigning frequency dependent weighting functions. Indeed,in [18], the algorithms are only used as an a posterioristability verification. The present paper aims to develop a data-driven iterative H∞ norm estimation procedure for model-error modeling of multivariable systems for robust control.

The contributions of this paper are threefold.C1) A novel multivariable data-driven H∞ norm estimation

procedure is presented that applies to general multi-inputmulti-output systems. In addition to iteratively determin-ing the worst-case frequency as in [15, Sec. 12.2], [16],and [17], the proposed algorithm in this paper is also ableto determine the worst-case input and output direction.

C2) The proposed multivariable data-drivenH∞ norm estima-tion procedure is used in conjunction with a new modeluncertainty structure that has recently been developed in[20]. This new model uncertainty structure renders the

x y

z

Fig. 1. Photograph of the experimental AVIS setup. The rotation around thex-axis is denoted by φ.

use of weighting functions superfluous in model-errormodeling by directly connecting the size of model uncer-tainty and the control criterion. This enables the use of theproposed data-driven H∞ norm estimation procedure formodel-error modeling of multivariable systems for highperformance robust control.

C3) Application of the procedure to a multivariable industrialAVIS system, which

i) confirms that the multivariable data-drivenH∞ normestimation procedure converges and performs reli-ably when implemented on an industrial system,

ii) reveals higher accuracy of the estimated model errorcompared to an alternative model-error modelingprocedure, and

iii) confirms that the resulting model set leads to en-hanced vibration isolation.

Preliminary research results related to C1 appear as [21],the present paper extends this with extensive theoretical andexperimental results. In addition, the present paper containsthe new contributions C2 and C3.

The main results of this paper are presented in the noise-freesituation to facilitate the exposition. An analysis of the situa-tion of additive stochastic noise is presented in Sec. III-C3.

The paper is organized as follows. In Sec. II, the AVIS andthe control goal are presented. Then, in Sec. III, the multivari-able data-driven H∞ norm estimation procedure is presented,leading to Contribution C1. Then, in Sec. IV, the proposedH∞ norm estimation procedure is used in conjunction withthe new model uncertainty structure in [20], constituting Con-tribution C2. In Sec. V experimental results of the proposeduncertainty modeling procedure are presented. The resultingmodel set is then used in Sec. VI to design and implementa robust controller on the AVIS system. Hence, Sec. V andSec. VI contain Contribution C3. Finally, a discussion and aconclusion are presented in Sec. VII.

II. DESCRIPTION OF AVIS AND CONTROL GOAL

A. AVIS

The AVIS in Fig. 1 is considered in this paper. The systemconsists of two main parts, i.e., a movable payload and achassis that is connected to the floor. The payload and chassis

Payloady

d1

d2

u

Fig. 2. Schematic illustration of the AVIS.

− C P

du y

Fig. 3. Simplified feedback interconnection for vibration isolation.

are connected by four isolator modules. On the one hand,these isolator modules provide passive damping through apneumatic airmount. On the other hand, the isolator modulesare equipped with Lorentz motors and geophones that enableactive vibration isolation. Specifically, the isolation modulesare each equipped with two motors, leading to eight actuatorsin total. The currents applied to the motors are denoteda =

[a1 a2 a3 a4 a5 a6 a7 a8

]. In addition, three

out of four modules are equipped with two geophones eachthat construct measurements of the velocity, leading to sixsensors, which are denoted s =

[s1 s2 s3 s4 s5 s6

].

The inputs and outputs of the system are rigid-body decoupledwith respect to the Cartesian coordinate frame depicted inFig. 1. The manipulated input and output are denoted

u = Tua (1)

y = Tys, (2)

respectively, where Tu ∈ R6×8 and Ty ∈ R6×6. Finally, thetrue system Po is given by Po : u 7→ y.

The goal of the AVIS is to isolate the payload with respect toexogenous disturbances. The AVIS is schematically illustratedin Fig. 2. The signal d1 refers to force disturbances thatdirectly affect the payload, and d2 refers to force disturbancesthat are induced by floor vibrations. The goal is to keep theabsolute velocity of the payload equal to zero by manipulatingthe input u through the velocity measurement y, see (2). Thekey advantage of active vibration control compared to thepassive isolation by the pneumatic airmounts is the availabilityof absolute velocity measurements. This enables skyhookdamping [3], which connects the fictive damper (implementedas a control algorithm) to the fixed world. As a result, skyhookdamping implies that y should be kept as small as possible forboth disturbances d1 and d2. This is in sharp contrast to thesituation where velocity measurements relative to the floor areavailable, in which case y should be kept as small as possibleonly if d2 = 0.

The resulting feedback interconnection is depicted in Fig. 3,where

d = d1 +Hdd2

10−1

100

101

102

−40

−30

−20

−10

0

10

20

30

40

50

60

f [Hz]

|.|[dB]

Fig. 4. Active vibration isolation through skyhook damping: identifiedfrequency response of open-loop Po in z-direction (solid blue) and φ-direction(dashed red) and identified frequency response of closed-loop P cl,exp

o in z-direction (dashed-dotted green) and φ-direction (dotted magenta).

and Hd is a causal stable transfer function matrix thatcharacterizes the transfer of the floor vibrations through thepneumatic airmount to the payload.

To show the potential performance improvement by skyhookdamping, an initial controller Cexp is designed. This controllerconsists of a multi-loop SISO controller, where each diagonalelement is a gain with high-frequency roll-off, see Fig. 14 inSec. VI-A for the diagonal elements corresponding to the zand φ directions. The resulting closed-loop transfer function

P cl,expo : d 7→ y = Po(I + CexpPo) (3)

characterizes the closed-loop disturbance attenuation proper-ties. In Fig. 4, frequency response function measurements ofthe open-loop system Po and the closed-loop system P cl,exp

o

are depicted in the vertical translational direction z. Theactual identification of these frequency response functions isdescribed in Sec. V. Clearly, Fig. 4, reveals that the controllerCexp damps the resonance phenomenon at approximately3.2 Hz by 10 dB in the translational z direction.

Although Cexp increases the damping properties of theAVIS, see Fig. 4, the increased damping is insufficient forsatisfactory control performance. This will be confirmed bythe experimental results in Sec. VI-B, specifically Fig. 16 andFig. 17. Analysis of the power spectral densities in Fig. 17reveals a dominant frequency content between 1 and 10 Hz.

B. Control goal and approach

The goal in this paper is to improve vibration isolation prop-erties of the AVIS through enhanced controller design. Thekey performance limiting factor in increasing the gain of thecontroller Cexp, and hence increasing the skyhook damping,involves the high-frequency flexible dynamics beyond 100 Hz,as can be observed in Fig. 4. As a result, performance androbustness objectives have to be appropriately specified. Thepursued approach to achieve high performance active vibrationisolation is a multivariable robust controller design.

The performance objectives are specified using a criterionJ (P,C), where the goal is to compute the optimal controller

Copt = arg minJ (Po, C).

Before specifying J (P,C) in more detail, see Sec. II-C, theconnection between the control objective and the true unknownsystem Po is established in further detail.

To address robustness objectives, note that the minimizationof J (Po, C) requires knowledge of Po. This knowledge isreflected by a model set P . This model set P will be definedmore precisely in Sec. II-D. The key property of this model setP is that it will be chosen such that it encompasses the trueAVIS dynamical behavior, i.e., including the high-frequencyflexible dynamics. Hence, the assumption

Po ∈ P (4)

is assumed to hold throughout the paper. The key motivationfor considering a model set is the fact that any model is un-certain and cannot represent the true system behavior exactly.In view of (4), given the model set P , the robust controllersynthesis

CRP = arg minCJWC(P, C), (5)

is considered, where JWC(P, C) = supP∈P J (P,C). Thisprovides a performance guarantee when implementing CRP

on the true system, since by (4), the bound

J (Po, CRP) ≤ JWC(P, CRP) (6)

holds.To facilitate the exposition, a two-input two-output robust

feedback controller is designed for the z and φ direction, i.e.,[uzuφ

]= CRP

[yzyφ

].

The presented approach applies equally well to the full mul-tivariable situation, i.e., three rotations and three translations.In this paper, the selection of a rotational and translationaldegree-of-freedom is made to show that the presented ap-proach automatically scales the uncertainty channels, see (28).Hence, it automatically deals with the various units of themeasurements. Note that all physical actuators a and sensors sare typically used due to Tu and Ty in (1) and (2), respectively.

C. Control criterion

The control goal in this paper is specified by

J = ‖WT (P,C)V ‖∞, (7)

where

T (P,C) =

[T11 T12

T21 T22

]=

[PI

](I + CP )−1

[C I

]. (8)

The transfer function matrix T (P,C) maps[r2

r1

]onto

[yu

],

where r1 corresponds to d in Fig. 3 and r2 is an additionalsignal, see Fig. 7 in Sec. IV-C for a block diagram. In addition,W and V in (7) are stable and minimum-phase weightingfilters. The motivation for considering the H∞ norm in (7) isto enable a systematic robust controller synthesis.

A four-block problem is considered, i.e., T (P,C) in (8)contains four blocks. The four-block problem guarantees in-ternal stability of the resulting optimal controller. This hasimportant implications from a theoretical perspective, since it

10−1

100

101

102

−70

−60

−50

−40

−30

−20

−10

0

10

f [Hz]

|.|[dB]

Fig. 5. Singular values of weighting filters W1 (solid blue), W2 (dashedred), and initial controller Cexp (dashed-dotted green).

will enable the construction of a specific coprime factorizationthat leads to (28). In addition, the four-block problem enablesthe use of the systematic loop-shaping approach in [22] toselect W and V such that it enhances active vibration isola-tion performance. Note that the relevant closed-loop transferfunction for vibration isolation is essentially given by (3).Interestingly, the loop-shaping approach in [22] is essentiallybased on the fact that the loop-gain CP is the only degree offreedom in shaping the four-block problem (8). In this paper,the transparent connection between shaping the loop-gain CPand the relevant transfer function (3) is exploited to design theweighting functions.

In the considered design framework, so-called loop-shapingweighting filters W1 and W2 are adopted that shape the desiredopen-loop gain W2PW1. Next, observe that the initial con-troller leads to a certain loop-gain CexpP . The desired loop-gain typically has a smaller amplitude in the low frequencyrange compared to W2PW1. Hence, the rationale in this paperto design the weighting filter W1 and W2 is to increase thegain of Cexp, i.e.,

W1 =

[1 00 1

], W2 =

[19.9 0

0 11.75

]Cexp.

Although the design procedure to select W2 resembles thecontroller Cexp, a robust controller synthesis procedure isrequired to deliver a robustly stabilizing feedback controller.These weighting functions W2 and W1 directly fit in criterion

(7) through W =

[W2 00 W−1

1

], V =

[W−1

2 00 W1

].

D. Identification of P using H∞ norm bounded perturbations

The uncertain model set P that is introduced in Sec. II-Bis constructed as an H∞ norm bounded perturbation around anominal model P , i.e.,

P ={P∣∣P = Fu(H(P ),∆u),∆u ∈∆u

}, (9)

where H(P ) represents the nominal model P and uncertaintystructure. Also, the upper linear fractional transformation(LFT) is given by

Fu(H,∆u) = H22 + H21∆u(I − H11∆u)−1H12.

As an example of (9), note that H(P ) in the case of additiveuncertainty P + ∆u is given by H11 = 0, H12 = H21 = I ,

and H22 = P . In addition, unstructured model uncertainty isconsidered, i.e.,

∆u := {∆u|‖∆u‖∞ ≤ γ} (10)

To actually identify P , three aspects are of importance:1) identifying a nominal model P ,2) constructing a model uncertainty structure, see H(P ), and3) determining the size of model uncertainty γ in (10).

In the next section, a new approach for the latter aspect,i.e., determining γ, is presented. The former two aspects aresubsequently addressed in Sec. IV, leading to a completeprocedure that encompasses 1)-3).

III. A DATA-DRIVEN APPROACH TO MULTIVARIABLEUNCERTAINTY MODELING

In this section, a novel procedure is presented to estimatethe size γ of ∆u, given a nominal model P and uncertaintystructure H(P ). As a result, γ depends on the choice of Pand the uncertainty structure, where a suitable structure willbe presented in Sec. IV. In addition, in view of (4), γ dependson the true system Po. To show this, let ∆o uniquely generatePo in (9), i.e.,

Po = Fu(H(P ),∆o). (11)

In this case,γ = ‖∆o‖∞

is the minimum-norm bound such that the model set definedby (9) and (10) satisfies (4). Thus, the goal in model-errormodeling essentially is to determine the H∞ norm of themodel error ∆o. Note that ∆o generally is a dynamic system,e.g., in the case of additive uncertainty, ∆o = P − Po.

The pursued approach to estimate γ is to perform experi-ments on ∆o. To achieve this, the input u∆ is applied to ∆o

and the output y∆ = ∆ou∆ is measured. Note that in generalexperimentation on ∆o involves both an experiment on the truesystem Po and a simulation with the model P . For instance, inthe case of additive uncertainty, y∆ = ∆ou∆ = P u∆−Pou∆.

The key idea in this paper is to directly estimate thesize γ of ∆o based on measured data, hence without theneed for identification of an intermediate model. In contrast,the approach in [7] involves the estimation of an auxiliarymodel of the model error ∆o, whereas [8] and [9] employa nonparametric model, i.e., an identified frequency responsefunction, of ∆o. In these approaches, the identified model issubsequently used to compute the H∞ norm.

Next, the basic procedure is presented, revealing that theauxiliary model estimation step can be rendered superfluous.

A. Iterative data-driven H∞ norm estimation: SISO systems

In this section the basic principle for data-driven H∞ normestimation is presented. To introduce the mechanism, firstassume that ∆ is SISO and linear time invariant (LTI). Hence,∆(z) =

∑∞k=0 δkz

−k, with δk, k = 0, . . . ,∞ the Markovparameters of the system. Next, recall that the H∞ norm is aninduced norm, i.e., ‖∆‖∞ = ‖∆‖i2 see [19, Appendix A.5.7]

for a proof. Assume that the signals have finite length N ∈ N,i.e., u∆, y∆ ∈ RN×1, hence

‖∆‖i2 = supu∆∈RN×1\0

‖y∆‖2‖u∆‖2

. (12)

Note that ‖∆‖i2 → ‖∆‖∞ for N →∞, see [17, Theorem 3]for a proof. Next, on the finite time interval of length N ,

y∆ = ∆u∆, (13)

where no measurement noise is assumed and

∆ =

δ0 0 0 . . . 0δ1 δ0 0 0...

.... . . 0

δN−1 δN−2 δN−3 . . . δ0

.

As a result, (12) is equivalent to

‖∆‖i2 = supu∆∈RN×1\0

√u∆

T∆T∆u∆

u∆Tu∆

=

√λmax(∆T∆).(14)

Now, observe that

∆T = TN∆TN , (15)

where

TN =

0 . . . 0 10 . . . 1 0...

...1 . . . 0 0

,

i.e., TN is an involutory permutation matrix of size N × N .Here, TN has the interpretation of a time-reversal operator.Next, (15) reveals that TN∆ is symmetric, hence

‖∆‖i2 = λmax(TN∆). (16)

As a result, ‖∆‖i2 equals the largest eigenvalue of the matrixTN∆. Hence, given the impulse response δi, i = 1, . . . , N ,‖∆‖i2 can directly be computed through an eigenvalue anal-ysis. In contrast, in this paper another approach is pursuedto determine ‖∆‖i2 that does not require knowledge of δi,i = 1, . . . , N . Such an approach is given next.

Procedure 1 (SISO ‖∆‖i2 estimation): Perform the fol-lowing sequence of steps.

1) set n = 1 and initialize with arbitrary u∆(1) ∈ RN×1\0.

2) determine y∆(n) = ∆u∆

(n) by performing an experimenton the system.

3) time-reverse: x∆(n) = TNy∆

(n).4) set u∆

(n+1) = x∆(n).

5) set n 7→ n + 1 and repeat from Step 2 until a stoppingcriterion is met.

Procedure 1 coincides with the power method [23, Sec. 7.3.1],which is an iterative algorithm to determine the maximaleigenvalue of a matrix. It is well-known that this estimator

γ(n)1 =

(u∆(n))

Tx∆

(n)

(u∆(n))

Tu∆

(n)(17)

converges under mild conditions to ‖∆‖i2 for n→∞.Essentially, two approaches can be pursued to complete

Step 2 in Procedure 1. On the one hand, ∆(n)u∆(n) can be

evaluated on a model using impulse response coefficients δi,

i = 1, . . . , N . On the other hand, the approach taken in thispaper is to perform an experiment of length N on the truesystem ∆o to evaluate ∆(n)u∆

(n). As a result, Procedure 1does not need any structural knowledge of ∆o.

Summarizing, the iterative procedure 1 leads to an estimateγ

(n)1 of ‖∆o‖∞ for a sufficiently large number of iterationsn and sufficiently long experiment length N . Each iterationrequires one experiment of length N on the true system ∆o,provided that ∆o is SISO. Since the approach is data-driven,the only computational step involves (17), which has linearcomplexity in N and thus is not restrictive for large N .

In the next section, H∞ norm estimation of multi-inputmulti-output (MIMO) systems is investigated.

B. Data-driven H∞ norm estimation of MIMO systems

In this section, a novel procedure for estimating the H∞norm of MIMO systems is presented. It turns out that Proce-dure 1 does not directly apply to MIMO systems, since (15) isnot valid for general MIMO systems. Hence, MIMO systemsintroduce a significant complication when compared to theSISO case in Sec. III-A.

To develop a data-driven H∞ norm estimator for MIMOsystems, consider the transfer function matrix of the MIMOsystem ∆ with p outputs and q inputs:

∆ =

∆11 . . . ∆1q

.... . .

...∆p1 . . . ∆pq

∈ RHp×q∞

with finite time representation for experiment length Ny∆1

...y∆p

︸ ︷︷ ︸=:y∆

=

∆11 . . . ∆1q

.... . .

...∆p1 . . . ∆pq

︸ ︷︷ ︸=:∆

u∆1

...u∆q

︸ ︷︷ ︸=:u∆

, (18)

where u∆i ∈ RN×1, i = 1, . . . , q and y∆i ∈ RN×1, i =1, . . . , p. In addition, ∆ ∈ RpN×qN .

To proceed, observe that (14) is also valid for the multivari-able system (18). Next, note that

∆T =

TN 0

. . .0 TN

︸ ︷︷ ︸=:TqN

∆11 . . . ∆p1

.... . .

...∆1q . . . ∆pq

︸ ︷︷ ︸=:∆

TN 0

. . .0 TN

︸ ︷︷ ︸=:TpN

,

i.e., ∆ is the finite time representation of the transfer functionmatrix

∆ =

∆11 . . . ∆p1

.... . .

...∆1q . . . ∆pq

.

In general, ∆ 6= ∆. As a result, an analogous step from (14)to (16) can only be made in the restrictive case where ∆ = ∆,i.e., if the system is symmetric. As a result, the case ∆ 6= ∆requires a different approach. The key step in the forthcomingdevelopments is the observation that

∆ =

q∑

i=1

p∑

j=1

Iij∆Iij , (19)

where

Iij =

0 . . . 0 . . . 0...

. . .. . . 0

0 . . . 1 . . . 0...

. . .. . .

...0 . . . 0 . . . 0

∈ Rq×p

and the 1 is on the (i, j)th location. To derive the finite timeequivalent of (19), note that the finite time representation ofIij is given by

Iij =

0 . . . 0 . . . 0...

. . .. . . 0

0 . . . IN . . . 0...

. . .. . .

...0 . . . 0 . . . 0

∈ RqN×pN , (20)

where Iij is a block-diagonal matrix of q × p blocks ofdimension N × N . Also, the (i, j)th block-element is equalto an identity matrix of size N . Note that the system Iij canbe interpreted as a static system, i.e., containing only a directfeedthrough term, leading to a diagonal matrix in (20). Next,

∆ =

q∑

i=1

p∑

j=1

Iij∆Iij . (21)

The result (21) is the basis for the following procedure,which constitutes Contribution C1 of this paper.

Procedure 2 (MIMO ‖∆‖i2 estimation): Perform the fol-lowing sequence of steps.

1) set n = 1 and initialize with arbitrary u∆(1) ∈ RqN×1,

u∆(1) 6= 0.

2) determine y∆(n) = ∆u∆

(n) by performing an experimenton the system.

3) time-reverse: x∆(n) = TpNy∆

(n).4) set z∆

(n) = 0 and perform pq experiments:for i = 1, ..., q

for j = 1, ..., pz∆

(n) 7→ z∆(n) + Iij∆Iijx∆

(n)

endend

5) time reverse: w∆(n) = TqNz∆

(n).6) set u∆

(n+1) = w∆(n).

7) set n 7→ n + 1 and repeat from Step 2 until a stoppingcriterion is met.

The multivariable H∞ norm can then be estimated as

γ(n)2 =

√(u∆

(n))Tw∆

(n)

(u∆(n))

Tu∆

(n), (22)

where in contrast to Procedure 1, a square root appears dueto the fact that the reduction from two experiments to a singleexperiment per step in the SISO procedure (based on (14) to(16)) is not made in the derivation of Procedure 2.

Basically, (19) and Procedure 2 recast the evaluation of ∆T

as p · q experiments on ∆, for which the true system ∆o isavailable for evaluation. Hence, the approach does not requireexplicit knowledge of ∆T and ∆. The underlying systemproperty that is exploited here is linearity. A single iterationof the proposed procedure 2 applied to a two-input two-outputMIMO system is illustrated in Fig. 6.

∆

TpN TqN∆

∆

∆

∆

∆

Exp. 1 Exp. 2 Exp. 3 Exp. 4

u∆ y∆

x∆ z∆ w∆

Step 2 Step 3 Step 4 Step 5

Fig. 6. Illustration of one iteration n of Procedure 2 for p = q = 2. Thedisconnected inputs are taken equal to zero. The different steps are indicatedin red. Step 4 involves four experiments that are explicitly indicated.

C. Convergence and implementation aspects

1) Convergence aspects: Procedure 1 and Procedure 2 areglobally convergent. The proof follows directly from the poweriterations method for computing the largest eigenvalue of amatrix, see [23, Sec. 8.2]. In addition, the finite time induced2 norm in (12) converges to the H∞ norm for N → ∞,see [17, Theorem 3]. Hence, N should be chosen sufficientlylarge to avoid transient effects. Such transient effects are well-known in system identification, e.g., in frequency domainsystem identification these are directly related to leakageerrors, see, e.g., [24]. Also, note that this result differs fromtime-domain model validation, including [13]. In particular,such model validation techniques also exploit the result (14)but are exact for both the infinite time and finite time case. Thekey difference lies in the fact that the power iteration approachin this paper involves an estimation problem instead of a modelvalidation problem. The key advantage of the power iterationsmethod is that very effectively deals with noise, as is describedin Sec. III-C3. In fact, in [17, Section 4.2], it is shown thatthe power iterations method can be interpreted as an iterativeexperiment design approach for estimating the H∞ norm.

2) Input constraints: The actual implementation on phys-ical systems of the data-driven H∞ norm estimation proce-dures 1 and 2 for SISO and MIMO systems, respectively,requires some additional attention. First, the input u∆

(n) tothe system ∆o is usually subject to constraints, including en-ergy, power, and amplitude constraints. These constraints candirectly be dealt with. In particular, let µ(n) be a normalizationconstant such that the input

1

µ(n)u∆

(n) (23)

satisfies the input constraints. Next, apply the scaled input in(23) to the system ∆o. Then, by linearity,

µ(n)∆o1

µ(n)u∆

(n) = ∆ou∆(n).

Hence, either the output of ∆o should be scaled by µ(n) or theestimators γ1 and γ2 should be appropriately extended withthe reciprocal of the normalization constant 1

µ(n) , i.e., µ(n).3) Noise aspects: In Sec. III-A and Sec. III-B, an iterative

estimation approach is presented that applies to the noise-free situation. A key aspect in any uncertainty modelling

procedure for robust control involves the distinction betweenexogenous noise and unmodelled dynamics. For instance,in model-validation approaches [12], [10], [13], it is testedwhether there exists an element in the model uncertainty classand disturbance signal that explain the measured data. Incontrast, the proposed power iteration method in Sec. III-A andSec. III-B involve an estimation problem. In the case whereadditive noise affects the measurements, i.e., (13) is extendedto y∆ = ∆u∆ + e, where e is zero-mean white noise withvariance λe, then two aspects are relevant.• if u∆ is assumed to converge to the eigenvector corre-

sponding to the largest eigenvalue λmax(TN∆), see (16)for the SISO case, then the estimator (17) is unbiased andits variance decreases with N and is proportional to λe,see [16, Section 3.1] for a proof.

• in [17], the convergence of the algorithm has beenestablished in the situation where noise is present. Inthis case, the signal u∆ may not converge to the desiredeigenvector. In fact, the noise affecting the output canbe interpreted as a re-initialization of the algorithm. Toavoid bias of the estimator γ(n)

1 , it has been argued in[17] that the iterative procedure should be extended toensure convergence.

Summarizing, the aspect of additive noise of the poweriteration algorithm has been extensively addressed in [16,Section 3.1],[17]. For the considered mechatronic applicationin this paper, the signal-to-noise ratio can be made sufficientlylarge by scaling the input appropriately using the resultsin Sec. III-C2. As a result, the input u∆ converges to theeigenvector corresponding to the largest eigenvalue and theestimators (17) and (22) are unbiased. In addition, due to agood signal to noise ratio and relatively long experiment, thevariance of the estimator is small.

IV. TOWARDS A MULTIVARIABLE MODELING PROCEDUREFOR ACHIEVING ROBUST PERFORMANCE

In the previous section, a novel approach has been presentedto estimate the H∞ norm γ of the model error in (10). As isargued in Sec. II-D, two aspects are remaining that determinethe shape of the model set P , i.e., 1) the nominal model P ,and 2) the uncertainty structure leading to H(P ) in (9).

The shape of the model set P clearly determines theworst-case performance bound in (6). In this section, a three-step procedure is proposed such that the nominal model P ,the uncertainty structure, i.e., H(P ), and the approach inSec. III-B to estimate γ jointly aim at achieving a small worst-case performance bound in (6). This is of crucial importancein view of the approach in Sec. III-B that only delivers a norm-bound. In contrast, alternative uncertainty modeling proce-dures often adopt weighting functions to reduce conservatism,e.g., [19, Sec. 7.4.3].

First, the general objective for modeling P is defined, afterwhich the three steps are described in detail. These three stepsconstitute Contribution C2 of the paper and are summarizedas follows.

Procedure 3: Perform the following three steps for identi-fying P:

1) Identify P in (25), see Sec. IV-B.2) Construct uncertainty model (27), leading to (28), see

Sec. IV-C.3) Apply Procedures 2 and 3, leading to γ and hence P in

(9) and (10).

A. Modeling goal

The function JWC(P, C) is a complex function of bothP and C. By noting that CRP in (5) depends on P , i.e.,CRP(P), it is desired to determine P such that it minimizesJWC(P, CRP(P)), subject to (4). However, this is in generaldifficult to solve.

The key step in this section is to exploit knowledge of Cexp,see Sec. II-A, to obtain a tractable approach that is aimed atachieving high performance in (5). Note that JWC(P, CRP) ≤JWC(P, Cexp). Hence, Cexp provides an upper bound for theguaranteed performance in (5). Hence, the aim is to determine

P = arg minJWC(P, Cexp)

subject to Po ∈ P.(24)

B. Step I: nominal modeling

In the first step, a nominal model P is identified. In view of(24), the control objective is also adopted in control-relevantidentification. In particular, the control-relevant identificationcriterion in [25] is adopted, i.e., P is minimized according to

minP

∥∥∥WT (Po, Cexp)V −WT (P , Cexp)V

∥∥∥∞. (25)

The actual minimization in (25) is performed using the ap-proach in [20] and is described in Sec. V-A. First, it is shownin Sec. IV-C that this criterion is useful in view of (24).

C. Step II: uncertainty model structure selection

Having identified a nominal model that minimizes (25), thenext step is the selection of an uncertainty model structure thatextends P to H(P ). As in Step 1, the essence lies in selectingthe uncertainty structure such that it facilitates solving (24).

To establish the connection between H(P ) and the criterion(24), note that the latter can be expressed as an LFT. Thisinvolves the construction of a generalized plant as is explainedin [19, Sec. 3.8]. In particular, the uncertain model P in (9) isappended with weighting filters and interconnected with Cexp,leading to the setup in Fig. 7. As a result,

JWC(P, Cexp) =sup∆u∈∆u

∥∥∥M22 + M21∆u(I − M11∆u)−1M12

∥∥∥∞.(26)

Since (26) depends on a complicated manner on ∆u andhence γ, see (10), a specific approach that relies on the resultsin [20] is adopted. In particular, in [20], it is suggested to

i) adopt the dual-Youla uncertainty structure [26], [27]

PDY =

{P |P =

(N +Dc∆u

)(D −Nc∆u

)−1,∆u ∈∆u

}, (27)

whereii) the pair {N , D} is a robust-control-relevant coprime

factorization of P as defined in [20, Sec. 3.3], and

HCexp−

∆u

WM

r1r2 yuV

Fig. 7. Worst-case performance JWC(P, Cexp) cast into the generalizedplant framework.

Cexp

−

r

yu

∆o

Nc Dc

D−1 N

Po

u∆

y∆

Fig. 8. Block diagram corresponding to uncertainty structure (27).

iii) the pair {Nc, Dc} is a (Wu,Wy)-normalized coprimefactorization of Cexp, see [20, Definition 4].

In (27), {N , D} is a coprime factorization of P if P = ND−1

and, in addition, N , D ∈ RH∞ and ∃X,Y such that XN +Y D = I . As a result of i), ii), and iii, (26) simplifies to

JWC(P, Cexp) ≤ ‖M22‖∞ + sup∆u∈∆u

‖∆u‖∞ = J (P , Cexp) + γ, (28)

see [20] for a proof of (28) and more details the specificcoprime factorizations.

The result (28) provides a direct connection between the sizeγ of ∆u, see (10), and the control criterion JWC(P, Cexp).Interestingly, the size of model uncertainty γ in (28) is equal tothe norm in the control-relevant identification objective (25).This implies that the same objective, i.e., (24), is being pursuedduring the nominal modeling step (Step I) and uncertaintymodeling step (Step II).

D. Step III: data-driven estimation of γ

Now, given H(P ), it remains to estimate γ to complete themodel set P . In this step, γ is estimated using the multivariabledata-driven H∞ norm estimation Procedure 2.

The key observation from Sec. III is that Procedure 2requires performing experiments on ∆o, i.e., the specificationof u∆ and measurement of y∆. The first step in this sectionis to investigate how to gain access to u∆ and y∆, giventhe uncertainty structure (27). As is described in Sec. III,the essential idea is to perform measurements on the ∆o todetermine its H∞ norm γ. Note that the ∆o that induces Poin (11) can be computed directly using (27), leading to

∆o = D−1c (I + PoC)−1(Po − P )D. (29)

The result (29) reveals that ∆o depends on the model P =ND−1, the true system Po, and Cexp = NcD

−1c . Hence, it is

expected that performing an experiment on ∆o both involvescomputations using the model P , as well as performingexperiments with Po.

To gain access to u∆ and y∆, note that (27) in closed-loop with Cexp implemented can be represented as in Fig. 8.Inspection reveals

u∆ = D−1(r − CexpNu∆ − CexpDcy∆ +Ncy∆),

implying that the reference signal

r = (D + CexpN)u∆. (30)

should be applied. Next, observe that

y∆ = D−1c (y − P (I + CexpP )−1r). (31)

Equations (30) and (31) reveal how experiments can be per-formed on ∆o. This is summarized in the following procedure.

Procedure 4 (Performing experiments on ∆o): Let inputu∆ be given and perform the following sequence of steps.1) Compute r = (D + CexpN)u∆.2) Perform a closed-loop experiment on Po with Cexp imple-

mented, i.e., y = Po(I + CexpPo)−1r

3) compute y∆ = D−1c (y − P (I + CexpP )−1r).

Procedure 4 can directly be implemented in Procedure 2, en-abling the data-driven estimation for multivariable uncertaintystructures given by (27).

Remark 1: Besides the fact that the specific coprime factorrepresentation used in (27) leads to the result (28), the generalrepresentation of (27) has important consequences for theanalysis of the algorithm in the presence of noise. Indeed,by the results presented in [28], the impact of noise thatenters the feedback loop in fact is equivalent to an open-loopidentification problem. Hence, the analysis of noise applied toopen-loop systems in [17] directly applies to the closed-loopresults presented in this paper.

V. EXPERIMENTAL RESULTS UNCERTAINTY MODELING

In this section, the multivariable modeling procedure pre-sented in Sec. IV is applied to the AVIS in Sec. II-A. Togetherwith Sec. VI, this constitutes Contribution C3 of the paper.

A. Step I. Nominal modeling

In the first step, the nominal model P in (25) is identified.A frequency response function of the closed-loop systemT (Po, C

exp) is identified using the approach in [24]. Thekey reason for the intermediate step of frequency responsefunction identification is that it enables the solution of (25)by exploiting the frequency domain interpretation of the H∞norm. By exploiting a multisine experiment design, the ap-proach in [24] enables the accurate identification of frequencyresponse functions by effectively reducing the variance errorwithout introducing bias. By virtue of (8), an estimate of Pcan be obtained from the relation P = T12T

−122 . The identified

frequency response function of Po is depicted in Fig. 9.Next, a model parameterization for P is used. An 8th order

model is used. The reason for this low order is that thecriterion (25) essentially shapes the bias of the parametricmodel. Hence, a low-order model suffices for control purposes.The actual optimization is performed using the algorithm in[20, Section 3.4-3.5]. The resulting parametric model P is

100

101

102

−50

0

50

|P|[dB],outputz input z

100

101

102

−50

0

50input φ

100

101

102

−180

−90

0

90

180

|P|[dB],outputφ

100

101

102

−180

−90

0

90

180

100

101

102

−50

0

50

6(P

)[◦],outputz

100

101

102

−50

0

50

100

101

102

−180

−90

0

90

180

6(P

)[◦],outputφ

f [Hz]10

010

110

2−180

−90

0

90

180

f [Hz]

Fig. 9. Nonparametric frequency response function Po (blue dots) andidentified parametric model P (dashed red). Both the z-translation and φ-rotation are displayed.

depicted in Fig. 9. In addition, the nonparametric frequencyresponse function of Po is depicted. Inspection reveals that thesuspended rigid-body mode at 4 [Hz] and the first resonanceat 135 [Hz] are accurately modeled.

B. Step II. Uncertainty model structure

Given the identified nominal model P in Step 1, the modeluncertainty structure in (27) is constructed. Importantly, therequired coprime factorizations in (27) follow directly fromthe pursued approach in [20], leading to the result (28).

To obtain an idea of the shape of ∆o, the frequency responsefunction of ∆o is identified. To achieve this, Procedure 4 isemployed in conjunction with the approach in [24, Chapter 2].It is emphasized that the resulting frequency response functionof ∆o is not required for using the power iterations algorithmin Sec. III. However, it does provide insight into the shape of∆o and a reference value for γ. In particular, the peak valueof σ(∆u) over frequency equals 1.95, which is attained ata frequency of 296 [Hz]. Note that a rather dense frequencygrid has been employed to determine the frequency responsefunction ∆o, leading to a fairly long experimentation time.

In the next section, Procedure 2 is applied to estimate γ.

C. Step III. Data-driven estimation of γ

In this section, the key experimental results of this paperare presented, which involve the application of Procedure 2 inview of uncertainty modeling of the AVIS.

The application of Procedure 2 leads to the followingobservations.

100

101

102

−60

−50

−40

−30

−20

−10

0

10

f [Hz]

σ(∆

o)[dB]

Fig. 10. Frequency response function estimate of ∆o.

1) The iteration is initialized with u∆(1) being zero mean

white noise, see Fig. 11 (Iteration 1) and Fig. 12 (Iteration1) for the corresponding power spectral density Ψ

u(1)∆

.

2) The output w∆(1) of ∆T

o ∆o in Fig. 11 (Iteration 1) is col-ored noise, since it contains several dominant frequencycomponents.

3) The resulting input u∆ and output w∆ in iteration n = 3are depicted in Fig. 11 (Iteration 3) and Fig. 12 (Iteration3). Both u∆ and w∆ contain several dominant frequencycomponents. In addition, note that the inputs during thedifferent iterations are normalized with respect to the2 norm. Interestingly, the output in iteration n = 3already has a significantly larger amplitude compared tothe output in iteration n = 1, see Fig. 11.

4) After 40 iterations, the input is mostly sinusoidal exceptfor truncation effects, see Fig. 11 and Fig. 12.

5) After convergence, u∆(40) is nonzero in both z-direction

and φ-direction, , see Fig. 11 and Fig. 12. Hence, Proce-dure 2 also determines the worst-case input direction.

Next, estimator (22) is evaluated for all 40 iterations, seeFig. 13. This leads to the following observations. First, theestimate γ(n)

2 converges for increasing n. This leads to

γ(40)2 = 1.997.

Second, when comparing the estimated γ(40)2 = 1.997 with the

result in Sec. V-B, which is based on the frequency responsefunction estimate in Fig. 10, it is observed that Procedure 2leads to a higher value γ

(40)2 compared to supω σ(∆o). In-

terestingly, inspection of Fig. 12 (Iteration 40) reveals thatthe dominant frequency component is 121 [Hz], which differssignificantly from the worst-case frequency of 296 [Hz] thatwas obtained in Sec. V-B. Hence, due to its guaranteed globalconvergence, Procedure 2 provides an effective means fordetermining the worst-case frequency in view of H∞ normestimation. This is in sharp contrast to the situation wherea fixed pre-chosen frequency grid is used, as is confirmedby the considered application. In this respect, Procedure 2 isconsidered as a useful adaptive experiment design approach.

D. Summary

The obtained nominal model P in Step I, together with theuncertainty structure in Step II, and the size γ(40)

2 = 1.997 inStep III, lead to a model set P , see (9). In the next section,the model set P is used to design a robust controller.

0 5 10

−0.2

0

0.2

Iteration1:u(1

)∆

z-direction

0 5 10

−0.2

0

0.2

φ-direction

0 5 10−0.5

0

0.5

Iteration1:w

(1)

∆

0 5 10−0.5

0

0.5

0 5 10

−0.2

0

0.2

Iteration3:u(3

)∆

0 5 10

−0.2

0

0.2

0 2 4 6 8−0.5

0

0.5

Iteration3:w

(3)

∆

0 5 10−0.5

0

0.5

0 5 10

−0.2

0

0.2

Iteration40:u(4

0)

∆

0 5 10

−0.2

0

0.2

0 5 10−0.5

0

0.5

t [s]

Iteration40:w

(40)

∆

0 5 10−0.5

0

0.5

t [s]

Fig. 11. Power iterations: iterations 1 (top), 3 (middle), 40 (bottom) ofProcedure 2 - measured inputs u∆ and outputs w∆.

0 100 200 300 400 50010

−10

10−5

100

Ψu(1)

∆

z-direction

0 100 200 300 400 50010

−10

10−5

100

φ-direction

0 100 200 300 400 50010

−10

10−5

100

Ψu(3)

∆

0 100 200 300 400 50010

−10

10−5

100

0 100 200 300 400 50010

−10

10−5

100

Ψu(40)

∆

f [Hz]0 100 200 300 400 500

10−10

10−5

100

f [Hz]

Fig. 12. Power iterations: iterations 1 (top), 3 (middle), 40 (bottom) ofProcedure 2 - power spectral density of the input u∆.

5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

Iteration n

γ(n

)2

Fig. 13. Estimated norm γ(n)2 .

TABLE IROBUST-CONTROL-RELEVANT IDENTIFICATION AND ROBUST

CONTROLLER SYNTHESIS RESULTS.

Controller Minimized criterion J (P , C) JWC(P, C)Cexp None 14.30 16.30CNP J (P , C) 1.87 ∞CRP JWC(P, C) 4.80 5.02

VI. CONTROLLER SYNTHESIS AND EXPERIMENTALIMPLEMENTATION

A. Robust controller synthesis

The model set P is used to analyze and synthesize severalcontrollers. In particular, the following controllers are consid-ered:• Cexp: initial controller that is described in Sec. II-A.• CNP: nominal controller CNP = arg minC J (P , C)• CRP: robust controller for model set P in (5).

The controllers CNP and CRP are computed using H∞-optimization and skewed-µ-synthesis, see [19] for details.

The controllers are depicted in Fig. 14, whereas the closed-loop process sensitivity functions P (I + CP )−1 are depictedin Fig. 15. In addition, the achieved performance for both themodel P and the model set P in terms of the control criterionare presented in Table I.

The following observations are made.1) When comparing J (P , Cexp) and JWC(P, Cexp), it is

observed that the bound (28) indeed holds and is tight.2) From Table I, CNP indeed achieves optimal performance

for the nominal model P . However, JWC(P, CNP) isunbounded. Hence, performance and stability cannot beguaranteed when implementing CNP on Po.

3) The controller CRP achieves the smallest worst-caseperformance, i.e., JWC(P, CRP). In addition, the per-formance for the nominal model, i.e., J (P , CRP) is alsosignificantly improved when compared to J (P , Cexp).

4) The improved performance in terms of the criterion co-incides with improved disturbance attenuation propertiesat low frequencies in Fig. 15, see also Sec. II-A.

B. Controller implementation

The synthesized controllers in Sec. VI-A are now imple-mented for validation. Measured time domain responses aredepicted in Fig. 16, whereas the corresponding cumulativepower spectral densities are depicted in Fig. 17.

The following observations are made. First, the controllerCNP does not stabilize the actual system. This is observedfrom the response in the φ-direction in Fig. 16, where the

100

101

102

−100

−50

0

|C|[dB],ou

tputy 1

input u1

100

101

102

−100

−50

0

input u2

100

101

102

−180

−90

0

90

180

6C

[◦],ou

tputy 1

100

101

102

−180

−90

0

90

180

100

101

102

−100

−50

0

|C|[dB],ou

tputy 2

100

101

102

−100

−50

0

100

101

102

−180

−90

0

90

180

6C

[◦],ou

tputy 2

f [Hz]10

010

110

2−180

−90

0

90

180

f [Hz]

Fig. 14. Controllers: Cexp (solid blue), CNP (dashed red), CRP (dashed-dotted green).

100

101

102

−20

0

20

40

60

|PS|[dB],ou

tputy 1

input u1

100

101

102

−20

0

20

40

60input u2

100

101

102

−180

−90

0

90

180

6PS

[◦],ou

tputy 1

100

101

102

−180

−90

0

90

180

100

101

102

−20

0

20

40

60

|PS|[dB],ou

tputy 2

100

101

102

−20

0

20

40

60

100

101

102

−180

−90

0

90

180

6PS

[◦],ou

tputy 2

f [Hz]10

010

110

2−180

−90

0

90

180

f [Hz]

Fig. 15. Closed-loop process sensitivities: P (I + CexpP )−1 (solid blue),P (I+CNPP )−1 (dashed red), P (I+CRPP )−1 (dashed-dotted green). Inaddition, the nominal model P is depicted (dotted magenta).

0 2 4 6 8 10−0.5

0

0.5

e(t)

z-direction

0 2 4 6 8 10−2

−1

0

1

2φ-direction

0 2 4 6 8 10−50

0

50

e(t)

0 2 4 6 8 10−50

0

50

0 2 4 6 8 10−0.5

0

0.5

t [s]

e(t)

0 2 4 6 8 10−2

−1

0

1

2

t [s]

Fig. 16. Measured time domain responses: Cexp (solid blue, top),CNP (dashed red, middle), CRP (dashed-dotted green, bottom). Here, low-frequency noise (normally distributed white noise, filtered through a low-passfilter with cut-off frequency of 8 Hz) is added to the system in both thez-translation and φ-rotation simultaneously.

10−1

100

101

102

0

0.005

0.01

0.015

0.02

0.025

f [Hz]

Ψe(t)

z-direction

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

f [Hz]

φ-direction

Fig. 17. Cumulative power spectral density (CPS) of the measured errorsignals in Fig. 16: Cexp (solid blue, top), CRP (dashed-dotted green, bottom).

system hits a safety guardrail due to the unstable behavior. Thiscorroborates the results in Table I, where JWC(P, CNP) isunbounded, and hence no performance and stability guaranteescan be given when implementing CNP on the true system Po.

Second, the experimental controller Cexp and optimal ro-bust controller CRP both stabilize the true system, which isrevealed by the stationary behavior in Fig. 16. In addition,the controller CRP leads to a significantly better performancewhen compared to Cexp, which is visible from the time do-main responses in Fig. 16 and especially from the cumulativepower spectral density in Fig. 17. In particular, the controllerCRP leads to a performance improvement of more than afactor 4 in z-translation and more than a factor 2 in φ-rotation.

VII. DISCUSSION AND CONCLUSION

A. Discussion

1) A model-error modeling perspective: From the perspec-tive of model-error modeling, the proposed approach does notrequire severe prior assumptions as in alternative approaches.Hence, it provides stronger guarantees in robust control design.

As is experimentally shown in Sec. V, Procedure 2 leadsto a higher and hence more accurate estimation of the H∞norm compared to the use of an a priori chosen frequency

grid, as is exemplified in Sec. V-B and also done in, e.g., [8].From this perspective, Procedure 2 can be seen as an adaptiveexperiment design procedure. As a result, Procedure 2 reducesestimation errors due to the use of a discrete frequency gridwhen compared to the uncertainty modeling procedure in [8].From this perspective, it is emphasized that the use of priorassumptions to bound the interpolation error, as is done in,e.g., [9], [10], is likely to lead to conservative results, as isalso argued in [11, 9.5.2].

2) Connections to related (iterative) approaches: The ap-proach presented in Sec. III is related to several identificationand iterative control algorithms. For a connection to maximumlikelihood estimation, see [17, Remark 10].

The iterative algorithm in Sec. III can be related to iterativelearning control (ILC) algorithms [29]. For simplicity, considerthe situation of Procedure 1, in which case the input duringthe next iteration is given by

u∆(n+1) = TNw∆

(n),

which closely resembles ILC update laws that are typicallyalso a linear function of the signals w∆

(n) and u∆(n). How-

ever, there are some principal differences when comparingtypical ILC algorithms and Procedure 1. To show this, assumeas in Sec. III-C that an input normalization is implemented1

as in (23), with µ(n+1) = ‖w∆(n)‖2, i.e., the 2 norm of the

input is normalized to 1, as is also done in [17]. This leads tothe input to the true system

u∆(n+1) = 1

‖w∆(n)‖2u∆

(n+1) = 1‖w∆

(n)‖2 TNw∆(n),

with output

w∆(n+1) = ∆ou∆

(n+1) = ∆o1

‖w∆(n)‖2 TNw∆

(n). (32)

In (32), the operator ∆o1

‖w∆(n)‖2 TN maps RN onto itself. Note

that the amplification of ∆o for the input w∆(n) equals the 2

norm of the output w∆(n+1). Under mild assumptions, which

are related to the comment in [17, Remark 9], ‖w∆(n+1)‖2 >

‖w∆(n)‖2. Since T has induced 2 norm equal to one, the

induced 2 norm of ∆o1

‖w∆(n)‖2 TN is larger than one. As a

result, (32) is a Lipschitz continuous function with Lipschitzconstant larger than one, hence (32) is not a contraction. Incontrast, ILC algorithms are also of the form (32), but designedsuch that the iteration (32) is a contraction map, correspondingto a Lipschitz constant that is strictly smaller than one. Thisensures convergence to a unique fixed point, generally beingan error signal of the system converging to zero.

The novel multivariable result of Procedure 2 providesfurther extensions to the implementation of ILC algorithms.In particular, in certain ILC algorithms referred to as adjointILC algorithms [30], it is desired to filter through the transposeof the system. It is clear that the use of time reversal operatorsas in (15) enables a direct evaluation on the true system andeffectively renders the need for a model in ILC algorithmssuperfluous. Interestingly, a slight modification of Procedure 2directly enables its use for multivariable ILC algorithms.

1As is argued in Sec. III-C, this requires a slight modification of theestimators (17) and (22), see also [17].

B. Conclusion

In this paper, a novel data-driven H∞ norm estimationprocedure, which relies on iterative experiments on the system,is proposed for multivariable systems. This procedure is em-ployed for model-error modeling purposes, and embedded ina system identification and robust control design framework.The key advantage of the proposed approach is that it providesguarantees to satisfy (4) without relying on severe priorassumptions that are adopted in alternative approaches. Ex-perimental results of an active vibration isolation system showfast convergence of the algorithm and its ability to determineboth the worst-case frequency and worst-case input and outputdirections in H∞ norm estimation. In addition, the experimen-tal results confirm its ability to provide enhanced vibrationisolation properties through a robust control design. Finally,the experimental results confirm that the iterative algorithmis useful for experiment design, i.e., to iteratively/adaptivelydetermine improved inputs for H∞ norm estimation, leadingto improved results compared to existing approaches.

The proposed results can be directly extended. Besidesits possible use in iterative learning control algorithms, seeSec. VII-A2, it can be directly used in any multivariableestimation problem involving eigenvalues. In addition, the pro-posed approach exhibits global convergence for multivariablesystems (see Section III-C). This requires an additional numberof experiments as is illustrated in Fig. 6. Present researchfocuses on reducing the number of required experiments.

ACKNOWLEDGEMENTS

The authors would like to thank Okko Bosgra, Robbert vanHerpen, and Bo Wahlberg for their contribution to this work.

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Tom Oomen received the M.Sc. degree (cum laude)and Ph.D. degree from the Eindhoven University ofTechnology, Eindhoven, The Netherlands, in 2005and 2010, respectively. He was awarded the CorusYoung Talent Graduation Award in 2005. He heldvisiting positions at the KTH, Stockholm, Swe-den and at The University of Newcastle, Australia.Presently, he is an assistant professor with the De-partment of Mechanical Engineering at the Eind-hoven University of Technology. He is an AssociateEditor on the IEEE Conference Editorial Board and

Guest Editor for IFAC Mechatronics. His research interests are in the field ofsystem identification and robust control, with applications in high-precisionmechanical servo systems.

Rick van der Maas received the M.Sc. degreein mechanical engineering in 2011 from the Eind-hoven University of Technology, Eindhoven, TheNetherlands. He is currently a Ph.D. candidate inthe Department of Mechanical Engineering at theEindhoven University of Technology, working incollaboration with Philips Healthcare. His researchinterests include system identification, modeling andcontrol with applications in high-precision and med-ical imaging systems.

Cristian R. Rojas was born in 1980. He received theM.S. degree in electronics engineering from the Uni-versidad Tecnica Federico Santa Marıa, Valparaıso,Chile, in 2004, and the Ph.D. degree in electricalengineering at The University of Newcastle, NSW,Australia, in 2008. Since October 2008, he has beenwith the Royal Institute of Technology, Stockholm,Sweden, where he is currently an Assistant Professorof the Automatic Control Lab, in the School ofElectrical Engineering. His research interests lie insystem identification and signal processing.

Hakan Hjalmarsson (M’98, SM’11, F’13) wasborn in 1962. He received the M.S. degree inElectrical Engineering in 1988, and the Licentiatedegree and the Ph.D. degree in Automatic Controlin 1990 and 1993, respectively, all from LinkpingUniversity, Sweden. He has held visiting researchpositions at California Institute of Technology, Lou-vain University and at the University of Newcastle,Australia. He has served as an Associate Editorfor Automatica (1996-2001), and IEEE Transactionson Automatic Control (2005-2007) and been Guest

Editor for European Journal of Control and Control Engineering Practice.He is Professor at the School of Electrical Engineering, KTH, Stockholm,Sweden. He is an IEEE Fellow and Chair of the IFAC Coordinating CommitteeCC1 Systems and Signals. In 2001 he received the KTH award for outstandingcontribution to undergraduate education. He is co-recipient of the EuropeanResearch Council advanced grant. His research interests include systemidentification, signal processing, control and estimation in communicationnetworks and automated tuning of controllers.