European Journal of Control 19 (2013) 279–288

Contents lists available at SciVerse ScienceDirect

European Journal of Control

0947-35http://d

☆ThisFoundat

n CorrE-m

zlatko.e

journal homepage: www.elsevier.com/locate/ejcon

H∞ gain-scheduled controller design for rejection of time-varyingnarrow-band disturbances applied to a benchmark problem$

Alireza Karimi n, Zlatko EmediLaboratoire d'Automatique, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

a r t i c l e i n f o

Article history:Received 10 January 2013Accepted 1 May 2013

Recommended by I.D. Landau/D.W. Clarke

the weighted infinity-norm of closed-loop transfer functions. This method is applied to a benchmark foradaptive rejection of multiple narrow-band disturbances. First, it is shown that a robust controller can be

Available online 10 May 2013

Keywords:Gain-scheduled controlRobust controlH∞ controlAdaptive disturbance rejectionActive suspension system

80/$ - see front matter & 2013 European Conx.doi.org/10.1016/j.ejcon.2013.05.010

research work is financially supported byion under Grant no. 200021-121749.esponding author. Tel.: +41 216933841; fax: +ail addresses: alireza.karimi@epfl.ch (A. Karimmedji@epfl.ch (Z. Emedi).

a b s t r a c t

A new method for H∞ gain-scheduled controller design by convex optimization is proposed that uses onlyfrequency-domain data. The method is based on loop shaping in the Nyquist diagramwith constraints on

designed for the rejection of a sinusoidal disturbance with known frequency. The disturbance model isfixed in the controller, based on the internal model principle, and the other controller parameters arecomputed by convex optimization to meet the constraints on the infinity-norm of sensitivity functions. Itis shown next that a gain scheduled-controller can be computed for a finite set of disturbancefrequencies by convex optimization. An adaptation algorithm is used to estimate the disturbancefrequency and adjusts the parameters of the internal model in the controller. The simulation andexperimental results show the good performance of the proposed control system.

& 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction

In control engineering problems, disturbance rejection is anextremely important task. Some disturbances have periodic char-acter and can even be expressed as combination of few sinusoidalsignals. Typical examples of systems with periodic disturbancesare hard disks [7], optical disk drives [21], helicopter rotor blades[2] and active noise control systems [16].

In the case that the disturbance frequency is known, certainapproaches, such as internal model control and repetitive controltechniques can be applied. If unknown frequency can be measureddirectly or indirectly, which happens e.g. in some active noisecontrol applications, adaptive feedforward control can be used forthe rejection of disturbance. In [4], it was shown that the standardadaptive feedforward control algorithm is equivalent to the inter-nal model control law. Survey on methods in both cases of knownand unknown disturbance frequency can be found in [5].

Since it is not always possible to measure the disturbancefrequency with a transducer, the parameter estimation methodsare often used to estimate the parameters of the disturbance model.Therefore, almost all unknown disturbance rejection algorithmsgenerally lead to a direct or indirect adaptive implementation, which

trol Association. Published by Elsev

the Swiss National Science

41 216932574.i),

can be referred to as “adaptive regulation”. The reason is that thecontroller parameters are adapted with respect to variations ofparameters of the disturbance model. In [17], two approaches arecompared on an active suspension system. The first approach is adirect adaptive control scheme with Q-parameterization of thecontroller, where the disturbance is rejected by adjusting theparameters of the Q polynomial. The second approach is an indirectadaptive control because the disturbance model is estimated firstand then, based on the internal model principle, new controller iscalculated to reject the disturbance.

A linear parameter-varying (LPV) controller design method isdescribed in [9] for rejection of sinusoidal disturbances. In thisapproach, an LPV controller is designed with H∞ performancebased on the method proposed in [11,1], using a single quadraticLyapunov function for all values of measured frequency. Discrete-time state-space state-feedback and full-order output feedbackLPV controller design methods are described in [13,3], guarantee-ing closed-loop stability for infinitely fast variations of disturbancefrequencies using a single Lyapunov matrix over the wholescheduling parameter space. A new method for fixed-order LPVcontroller design with application to disturbance rejection of anactive suspension system is proposed in [10].

In this paper, a fixed-order H∞ gain-scheduled controller designmethod based only on the frequency-domain data is proposed. Inthis method, computation of the controller parameters and theirinterpolation are performed by one convex optimization. More-over, a solution to a challenging benchmark problem [19] forrejection of time-varying narrow-band disturbances is provided.

ier Ltd. All rights reserved.

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288280

The results are computed using a new public-domain toolboxfor robust controller design in the frequency domain which isavailable online [15].

The advantages of the proposed method with respect to theLPV controller design methods are

�

There is no need for a parametric model of the plant and thefrequency response can be used directly for controller design.As a result, this approach can be used for discrete- continuous-time systems with pure time delay.

�

The controller order is fixed, which allows less computationeffort in real-time applications.

�

Since the stability and performance are guaranteed for frozenscheduling parameters, the method has less conservatism withrespect to LPV controllers based on quadratic stability.

The last item can be considered as a drawback as well, because thestability and performance are not guaranteed for fast variations ofthe scheduling parameters. The other drawbacks are the limitationof the controller structure to linearly parameterized controllersand the dependence of the final solution on the choice of thedesired open-loop transfer function Ld. Some propositions toovercome these drawbacks are given in [14].

The proposed method, like the other adaptive methods used inthe benchmark, guarantees the stability and performance only forfrozen and slowly time-varying scheduling parameters. However, inpractice, as it is shown for the benchmark problem, the designedcontrollers stabilize the system even for rather fast variations ofdisturbance frequencies.

The paper is organized as follows: Section 2 describes the gain-scheduled H∞ controller design method that uses only the fre-quency response of the model. The method is applied to thebenchmark problem for adaptive disturbance rejection of an activesuspension system in Section 3. Section 4 presents the simulationand the experimental results. Finally, Section 5 gives some con-cluding remarks.

2. Gain-scheduled H∞ controller design

A fixed-order H∞ controller design method for spectral modelsis proposed in [14]. In this section we extend this method to designof gain-scheduled H∞ controllers.

A classical way to design gain-scheduled controllers includestwo steps:

1.

A set of controllers, one per operating point, is designed (if theoperating points are a continuous function of a schedulingparameter, a fine grid is used to obtain a finite set).

2.

The controller parameters are interpolated by a polynomialfunction of the scheduling parameter.

In order to reduce the complexity of the gain-scheduled controller, alinear or low-order interpolation is normally used. In this case, thestability and performance are not necessarily preserved even for thegridded scheduling parameter. The method that we propose putsthese two steps together and computes a gain-scheduled controllerthat satisfies the stability and H∞ performance conditions for allgridded values of the scheduling parameter using the convexoptimization methods. For the ease of presentation, a scalar sche-duling parameter and one H∞ constraint on the weighted sensitivityfunction are considered. The extension to vector of schedulingparameters and H∞ constraints on several sensitivity functions isstraightforward. In the sequel, the class of models, controllers anddesign specifications are defined and a convex optimization problemis proposed that results in a gain-scheduled controller.

2.1. Class of models

The class of causal discrete-time LTI-SISO models with boundedinfinity-norm is considered. It is assumed that the spectral model ofthe system as a function of the scheduling parameter θ, Gðe−jω; θÞ isavailable. The bounded infinity norm condition will be relaxed lateron to consider systems with poles on the unit circle. Since only thefrequency-domain data are used in the design method the exten-sion to continuous-time systems is straightforward (see [14]).

2.2. Class of controllers

Linearly parameterized discrete-time gain-scheduled control-lers are considered:

Kðz−1; ρðθÞÞ ¼ ρT ðθÞϕðz−1Þ; ð1Þwhere

ϕT ðz−1Þ ¼ ½ϕ1ðz−1Þ;ϕ2ðz−1Þ;…;ϕnðz−1Þ� ð2Þrepresents the vector of n stable transfer functions, namely basisfunctions vector that may be chosen from a set of generalizedorthonormal basis functions, e.g. Laguerre basis [20], and

ρT ðθÞ ¼ ½ρ1ðθÞ; ρ2ðθÞ;…; ρnðθÞ� ð3Þrepresents the vector of controller parameters. The dependence ofthe controller parameters ρi to θ can be affine or polynomial, e.g.ρiðθÞ ¼ ρi0 þ ρi1θ þ⋯þ ρinθ θ

nθ .The main reason to use a linearly parameterized controller is

that every point on the Nyquist diagram of the open-loop transferfunction becomes a linear function of the vector of controllerparameters ρðθÞ:Lðe−jω; ρðθÞÞ ¼ Kðe−jω; ρðθÞÞGðe−jω; θÞ ð4Þ

Lðe−jω; ρðθÞÞ ¼ ρT ðθÞϕðe−jωÞGðe−jω; θÞ; ð5Þthat helps obtaining a convex parameterization of fixed-order H∞

controllers.

2.3. Design specifications

The nominal performance can be defined by (see [8])

∥W1SðρðθÞÞ∥∞o1 ∀θ; ð6Þwhere Sðz−1; ρðθÞÞ ¼ ½1þ Lðz−1; ρðθÞÞ�−1 is the sensitivity function andW1 represents the performance weighting filter. The approachproposed in [14] is based on the linearization of this constraintaround a known desired open-loop transfer function Ld (that may bea function of θ as well). The main interest of this linearization is thatit gives not only sufficient conditions for the nominal performancebut also some conditions on Ld that guarantee the stability of theclosed-loop system. The linear constraints are given by [14]

jW1ðe−jωÞ½1þ Ldðe−jω; θÞ�j−Ref½1þ Ldðejω; θÞ�½1þ Lðe−jω; ρðθÞÞ�go0; ∀ω; ∀θ ð7Þ

It is easy to show that the inequality in (6) is met if the aboveinequality is satisfied. Knowing that the real value of a complexnumber is always less than or equal to its absolute value, we have

jW1ðe−jωÞ½1þ Ldðe−jω; θÞ�j−j1þ Ldðejω; θÞjj1þ Lðe−jω; ρðθÞÞjo0; ∀ω; ∀θ ð8Þ

which leads to

jW1ðe−jωÞjo j1þ Lðe−jω; ρðθÞÞj; ∀ω; ∀θ ð9Þthat is equivalent to (6). Moreover, it can be shown that the numberof encirclements of the critical point by L and Ld is equal. As a result,the closed-loop stability is ensured if LdðθÞ satisfies the Nyquist

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288 281

criterion for all θ (e.g. it does not turn around −1 for stable plantmodels). On the other hand, if the plant model and/or the controllerhave unbounded infinity-norm, i.e. the poles on the unit circle, thesepoles should be included in Ld (see [14]).

The graphical interpretation of this method is given in Fig. 1. Itis well known that the H∞ performance condition in (6) is satisfiedif and only if there is no intersection between Lðe−jω; ρðθÞÞ and acircle centered at −1 with radius jW1ðejωÞj [8]. It is clear that thiscondition is satisfied if Lðe−jω; ρðθÞÞ lies at the side of d that excludes−1 for all ω and θ, where d is tangent to the circle and orthogonalto the line connecting −1 to Ldðe−jω; θÞ. The conservatism of theproposed approach depends on the choice of Ld and is discussed in[14]. It is clear that if Ld ¼ L there is no conservatism. Therefore,choosing Ld as close as possible to L will reduce significantly thisconservatism. Since L is not a priori known, an iterative approachcan be used to reduce the conservatism (at each iteration L of theprevious iteration is used as Ld).

2.4. Optimization problem

The constraints in (7) should be satisfied for all ω∈½0;ωn�, whereωn is the Nyquist frequency, and for all θ∈½θmin; θmax�. This leads to aninfinite number of constraints that is numerically intractable. Apractical approach is to choose finite grids for ω and the schedulingparameter θ and find a feasible solution for the grid points. Thisleads to a large number of linear constraints that can be handledefficiently by linear programming solvers. By increasing the numberof scheduling parameters, the number of constraints will increasedrastically that increases the optimization time. In this case ascenario approach can be used to guarantee the satisfaction of allconstraints with a probability level when they are only satisfied for afinite number of randomly chosen scheduling parameter values [6].Some of the effects of gridding in frequency and additional con-straints that can be imposed for ensuring good behavior betweenthe grid points are described in [12].

Fig. 1. Linear constraints for robust performance.

Fig. 2. Block diagram of the active suspension system.

3. Active suspension benchmark

The objective of the benchmark is to design a controller for therejection of unknown/time-varying multiple narrow band distur-bances located in a given frequency region. The proposed con-trollers will be applied to the active suspension system ofthe Control Systems Department in Grenoble (GIPSA–lab) [19].The block diagram of the active suspension system together withthe proposed gain scheduled controller is shown in Fig. 2.

The system is excited by a sinusoidal disturbance v1ðtÞ gener-ated using a computer-controlled shaker, which can be repre-sented as a white noise signal, e(t), filtered through thedisturbance model H. The transfer function G1 between thedisturbance input and the residual force in open-loop, yp(t), iscalled the primary path. The signal y(t) is a measured voltage,representing the residual force, affected by the measurementnoise. The secondary path is the transfer function G2 betweenthe output of the controller u(t) and the residual force in open-loop. The control input drives an inertial actuator through a poweramplifier. The sampling frequency for both identification andcontrol is 800 Hz, as chosen by the benchmark organisers.

The disturbance consists of one to three sinusoids, leading to threelevels of benchmark depending on the number of sinusoids. Dis-turbance frequencies are unknown but lie in an interval from 50 to95 Hz. The controller should reject the disturbance as fast as possible.We explain in detail the control structure and the design method forLevel 1. The extension to the other levels is straightforward.

3.1. Controller design for Level 1

An H∞ gain-scheduled controller, based on the internal modelprinciple to reject the disturbances, is considered as follows:

Kðz−1; θÞ ¼ ½K0ðz−1Þ þ θK1ðz−1Þ�Mðz−1; θÞ ð10Þwhere K0 and K1 are FIR filters of order n and

Mðz−1; θÞ ¼ 11þ θz−1 þ z−2

ð11Þ

the disturbance model for a sinusoidal disturbance with frequencyf 1 ¼ cos−1ð−θ=2Þ=2π. In order to improve the transient response, theinfinity norm of the transfer function between the disturbance andthe output, MG1S, should be minimized. However, since the primarypath model G1 cannot be used in the benchmark, it is replaced by aconstant gain. On the other hand, in order to increase the robustnessand prevent the activity of the command input at frequencies wherethe gain of the secondary path is low, the infinity norm of the inputsensitivity function ∥KS∥∞ should be decreased as well. Anotherconstraint on the maximum of the modulus of the sensitivityfunction ∥S∥∞o2 (6 dB) is also considered according to the bench-mark requirements (not to amplify the noise at other frequencies).

A gain-scheduled controller is designed using the followingsteps:

1.

Because of very high resonance modes in the secondary pathmodel, a very fine frequency grid with a resolution of 0.5 rad/s(5027 frequency points) is considered.

2.

The interval of the disturbance frequencies is divided to 46points (a resolution of 1 Hz), which corresponds to 46 points inthe interval ½−1:8478;−1:4686� for the scheduling parameter θ.

3.

The following optimization problem is solved:

min γ

γ−1½jMðe−jωk ; θiÞj þ jKðe−jωk ; ρðθiÞÞj� � ½1þ Ldðe−jωk ; θiÞ�j−Ref½1þ Ldðejωk ; θiÞ�½1þ Lðe−jωk ; ρðθiÞÞ�go0;

−5

0

5

10

15

de (d

B)

Bode Diagram

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288282

0:5j½1þ Ldðe−jωk ; θiÞ�j−Ref½1þ Ldðejωk ; θiÞ�½1þ Lðe−jωk ; ρðθiÞÞ�go0for k¼ 1;…;5027; i¼ 1;…;46 ð12Þ

where the first constraint is the convexification of ∥jMSj þjKSj∥∞oγ and the second constraint that of ∥S∥∞o2. This is aconvex optimization problem for fixed γ and can be solved byan iterative bisection algorithm.

Remarks.

−10agni

tu

�

Mag

nitu

de (d

B)

Figcies

−20

−15

M

The controller order (the order of the FIR models for K0 and K1

in (10)) is chosen equal to 10 (the controller order is increasedgradually to obtain acceptable results). Note that it is much lessthan the order of the plant model, which is equal to 26.

�

−30

−25

The desired open-loop transfer functions are chosen asLdðθiÞ ¼ KiniðθiÞG2, where KiniðθiÞ are stabilizing controllers com-puted by pole placement technique.

50 100 150 200 250 300 350

� Frequency (Hz)

Fig. 4. Magnitude plot of the input sensitivity functions for disturbance frequencies

The Frequency-Domain Robust Control Toolbox [15] is used forsolving this problem. For the convenience, the internal model isconsidered as a part of the plant model, i.e. GðθÞ ¼MðθÞG2, andafter the controller design it is returned to the controller.

from 50 Hz to 95 Hz.

� After 7 iterations of the bisection algorithm γmin ¼ 1:68 isobtained. The total computation time is about 11 min on apersonal computer (16GB of DDR3 RAM memory at 1600 MHzand processor Intel Core i7 running at 3.4 GHz).

The parameters of the final designed gain-scheduled controllerin (10) are given by

K0ðz−1Þ ¼ 3:669−2:311z−1−0:7776z−2 þ 0:7171z−3

þ3:424z−4−5:402z−5 þ 5:077z−6−5:143z−7 þ 4:637z−8

−2:01z−9 þ 0:5125z−10

K1ðz−1Þ ¼ 2:241−1:293z−1−0:7633z−2 þ 0:4309z−3

þ2:673z−4−3:921z−5 þ 3:117z−6−2:638z−7 þ 2:476z−8

−1:15z−9 þ 0:3444z−10

This gain-scheduled controller gives very good transient perfor-mance and satisfies the constraint on the maximum modulus ofthe sensitivity function for all values of the scheduling parameter.Figs. 3 and 4 show the magnitude of the output sensitivity

0 50 100 150 200 250 300 350 400−20

−15

−10

−5

0

5

10Bode Diagram

Frequency (Hz)

. 3. Magnitude plot of the output sensitivity functions for disturbance frequen-from 50 Hz to 95 Hz.

functions S and the input sensitivity functions KS, respectively,for 46 gridded values of the disturbance frequencies. One canobserve very good attenuation at the disturbance frequencies andthe satisfaction of the modulus margin of at least 6 dB for alldisturbances.

3.2. Controller design for Level 2

In this level of the benchmark, two sinusoidal disturbancesshould be rejected. The structure of the gain scheduled controlleris given by (z−1 is omitted)

Kðθ1; θ2Þ ¼ ðK0 þ θ1K1 þ θ2K2ÞMðθ1; θ2Þ ð13Þwhere K0;K1 and K2 are 8th order FIR filters and

Mðθ1; θ2Þ ¼1

1þ θ1z−1 þ θ2z−2 þ θ1z−3 þ z−4ð14Þ

By considering a hard constraint on the magnitude of the sensi-tivity function ∥ð1þ KG2Þ−1∥∞o2:24 (7 dB) the optimizationbecomes infeasible. Therefore, the following constraint is consid-ered for optimization:

jMð1þ KG2Þ−1j þ jð1þ KG2Þ−1joγ ∀ω; ∀θ1; ∀θ2 ð15Þwhere γ is minimized. The first term on the left hand siderepresents the approximation of the disturbance path impulseresponse (ignoring the unknown transfer function G1). By mini-mizing its ∞�norm, we indirectly reduce the transient time (with atradeoff between fast response and robustness guarantee, comingfrom the second term).

Since we have two scheduling parameters a resolution of 1 Hzfor each sinusoidal disturbances leads to 462=2¼ 1058 grid points.This increases by a factor of 23 the number of constraints withrespect to that of Level 1. Moreover the resolution of the frequencygrid is improved from 0.5 rad/s to 0.2 rad/s which increases thenumber of constraints. The number of variables is also increasedfrom 22 (the coefficients of two FIR of order 10) to 27 (thecoefficients of three FIR of order 8). In order to solve theoptimization more quickly, the scenario approach is used. Fromthe set of 1058 frequency pairs, 50 samples are chosen randomlyand the constraints are considered just for these frequencies. Thestability of the closed-loop system however, is verified a posteriorifor all 1058 frequency pairs, which makes the probability ofstability constraint violation (between the grid points) very low.

0 50 100 150 200 250 300 350 400−20

−15

−10

−5

0

5

10

15

20

Mag

nitu

de (d

B)

Bode Diagram

Frequency (Hz)

Fig. 6. Magnitude of the input sensitivity functions for known disturbancefrequencies in F .

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288 283

The computed controller, however, destabilized the real system fordisturbance frequency pair (50–70) Hz. The main reason is themodeling error for the secondary path model around 50 Hz.Therefore, a new model for the secondary path provided by thebenchmark organizers with smaller modeling error around 50 Hzis used for the controller design. A new controller is designedusing the scenario approach and achieves γmin ¼ 10:62 after 11iterations with a total computation time of about 15 min.

The controller parameters are

K0ðz−1Þ ¼−4:835þ 43:93z−1−98:74z−2 þ 96:2z−3

−0:3158z−4−100:8z−5 þ 117:5z−6−64:79z−7 þ 16:35z−8

K1ðz−1Þ ¼−24:02þ 122:4z−1−226:7z−2 þ 204:9z−3

−52:21z−4−71:05z−5 þ 80:54z−6−36:53z−7 þ 8:671z−8

K2ðz−1Þ ¼−16:05þ 77:44z−1−139:6z−2 þ 124:7z−3

−37:19z−4−28:47z−5 þ 31:64z−6−11:83z−7 þ 2:561z−8

Figs. 5 and 6 show the magnitude of the output and inputsensitivity functions for known disturbance frequencies taken fromthe following set:

F ¼ fð50;70Þ; ð55;75Þ; ð60;80Þ; ð65;85Þ; ð70;90Þ; ð75;95ÞgThe attenuation of at least 40 dB is obtained for all frequencies butthe maximum of the output sensitivity function is greater than7 dB in some frequencies.

3.3. Controller design for Level 3

Although very good results can be obtained for linear controllerdesign for every triplet disturbance frequencies, a simple gain-scheduled controller that satisfies all constraints could not byobtained by the proposed approach. In fact the optimizationproblem becomes infeasible for affine dependence of the control-ler parameters to the scheduling ones and with relaxing theconstraints the resulting stabilizing controller does not lead togood performance even for known disturbance frequencies.

0 50 100 150 2−30

−25

−20

−15

−10

−5

0

5

10

15

20

Mag

nitu

de (d

B)

Bode

Freque

Fig. 5. Magnitude of the output sensitivity functi

It is important to emphasize that there is no theoreticallimitation for having three or even more disturbance frequencies.However, increasing the number of frequencies increases thecomplexity of the optimization problem such that a feasiblesolution could not be found in the first trials. This problem couldbe fixed by designing better initial controllers for each fixedfrequency and using these controllers for computing Ld, as wellby the better choice of the basis functions. However, because ofthe deadline for the benchmark, the authors decided just toparticipate in the first and the second level.

3.4. Estimator design

The scheduling parameter θ used in the internal model ofdisturbance in (11) is estimated using a parameter adaptationalgorithm. To estimate the parameters of the disturbance model,we need to measure the disturbance signal p(t) (see Fig. 2). If wemodel p(t) as the output of an ARMA model with white noise asinput, we have

Dpðq−1ÞpðtÞ ¼Npðq−1ÞeðtÞ; ð16Þ

00 250 300 350 400

Diagram

ncy (Hz)

ons for known disturbance frequencies in F .

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288284

where e(t) is a zero mean white noise with unknown variance.Estimation of the parameters of Np and Dp could be performed bythe standard Recursive Extended Least Squares method [18], if p(t)was measured. Since p(t) is not available, it is estimated using themeasured signal y(t) and the known model of the secondary path.From Fig. 2, we have

pðtÞ ¼ yðtÞ−q−dBðq−1ÞAðq−1Þ uðtÞ−v2ðtÞ; ð17Þ

where q−dBðq−1Þ=Aðq−1Þ is the parametric model of the secondarypath G2. Since v2ðtÞ is a zero mean noise signal, unbiased estimateof p(t) is given as

pðtÞ ¼ yðtÞ þ ½Aðq−1Þ−1�½yðtÞ−pðtÞ�−Bðq−1Þuðt−dÞ

For the asymptotical rejection of sinusoidal disturbance, there isno need to identify the whole model of the disturbance path, i.e.HG1 as shown in Fig. 2. The information needed is just thefrequency of the disturbance. So, by setting Dpðq−1; θÞ ¼ 1−θq−1 þq−2 (for Level 1) and Npðq−1Þ ¼ 1þ c1q−1 þ c2q−2, a simple para-meter estimation algorithm can be developed. Let us define

zðt þ 1Þ ¼ pðt þ 1Þ þ pðt−1Þ ð18Þ

ψT ðtÞ ¼ ½−pðtÞ; εðtÞ; εðt−1Þ�T ð19Þ

ΘT ðtÞ ¼ ½θ; c1; c2�T ð20Þ

where εðtÞ ¼ zðtÞ−zðtÞ is the a posteriori prediction error. Now, thefollowing recursive adaptation algorithm can be used to estimatethe scheduling parameter θ:

ε3ðt þ 1Þ ¼ zðt þ 1Þ−ΘðtÞψ ðtÞ

εðt þ 1Þ ¼ ε3ðt þ 1Þ1þ ψT

f ðtÞFðtÞψ f ðtÞ

Θðt þ 1Þ ¼ ΘðtÞ þ FðtÞψ f ðtÞεðt þ 1Þ

Table 1Simple step test (simulation)—Level 1.

Frequency (Hz) Global (dB) Dist. atte. (dB) Max. amp. (dB at Hz) N

50 30.0539 22.1214 9.6674 at 53.12 1355 33.0839 39.1389 5.7785 at 114.06 1160 32.9298 40.6649 5.2791 at 78.12 965 33.1775 39.2777 5.4087 at 73.43 770 33.5947 47.4173 4.6449 at 51.56 775 34.2959 42.8627 3.6597 at 50.00 880 34.8302 45.3628 3.8744 at 50.00 885 34.5090 43.2440 4.0071 at 50.00 890 32.3077 39.3682 4.4651 at 50.00 1295 23.9978 23.8150 4.9679 at 50.00 15

Table 2Simple step test (experimental results)—Level 1.

Frequency (Hz) Global (dB) Dist. atte. (dB) Max. amp. (dB at Hz) N

50 32.1963 26.0390 13.19 at 117.19 2855 32.9624 41.5091 11.66 at 125.00 1360 33.7955 41.3196 11.59 at 70.31 965 32.5293 45.4435 9.54 at 134.37 970 30.0156 42.6926 11.41 at 134.37 975 30.9359 43.1902 9.74 at 137.50 780 29.6325 44.9083 9.43 at 137.50 885 28.3826 38.3824 7.63 at 118.75 890 28.2388 37.0264 10.02 at 135.94 895 28.8061 37.0992 7.36 at 114.06 8

Fðt þ 1Þ ¼ 1λ1ðtÞ

FðtÞ−FðtÞψT

f ðtÞψ f ðtÞFðtÞλ1ðtÞλ2ðtÞ

þ ψTf ðtÞFðtÞψ f ðtÞ

2664

3775 ð21Þ

where ψ f ðtÞ ¼ 1=ðNpðq−1ÞÞψðtÞ, ε○ðtÞ is the a priori prediction errorand λ1ðtÞ and λ2ðtÞ define the variation profile of the adaptation gainF(t). Filtered observation vector ψ f ðtÞ is used to ensure the stabilityand convergence properties of the adaptation algorithm ([18]). Theother condition for the convergence, namely the richness of excita-tion, is satisfied as long as disturbance is not zero. A constant tracealgorithm [18] is used for the adaptation gain.

The same recursive adaptation algorithm is used for Level 2 ofthe benchmark with the difference that the order of the distur-bance model and consequently the number of the schedulingparameters is increased (θ is replaced by a vector ½θ1; θ2�).

4. Simulation and experimental results

The simulation results are presented for three different tests ofeach benchmark level: simple step test, step changes in frequen-cies test and chirp test, according to the benchmark requirements.

4.1. Simple step test

The simulation and experimental results for Level 1 are given inTables 1 and 2, respectively. The first column gives the globalattenuation in dB. It is the ratio of the energy of the disturbance inopen-loop to that in closed-loop computed in steady state (lastthree seconds of the experiment). The second column shows theattenuation at the disturbance frequency. The maximum amplifica-tion of the disturbance at other frequencies is computed and shownin the third column together with the frequency at which it occurs.The two-norm of the transient response of the residual force isgiven in the fourth column and the two norm at the steady state(last three seconds) in the fifth column. The peak value of the

orm2 trans. (�10−3) Norm2 res. (�10−3) Max. val. (�10−3) BSI (%)

.4803 6.0275 19.2347 100.00

.2054 4.2825 21.5834 100.00

.1452 4.3614 21.5092 100.00

.9451 4.3065 19.5405 100.00

.9827 4.1534 22.6636 100.00

.1354 3.9172 22.5296 100.00

.0156 3.6393 21.3056 100.00

.6001 3.6477 23.4386 100.00

.5930 3.8676 25.5074 100.00

.5999 4.4858 29.8633 100.00

orm2 trans. (�10−3) Norm2 res. (�10−3) Max. val. (�10−3) BSI (%)

.7275 9.8031 22.2959 83.88

.7586 5.6248 18.5939 100.00

.9979 5.1623 17.3711 97.99

.8304 5.0178 19.3765 100.00

.3400 5.5506 20.6127 95.06

.7819 4.4682 15.7354 100.00

.5284 5.0297 21.8171 100.00

.0995 5.7268 20.5997 100.00

.8059 5.0778 23.0987 100.00

.5047 4.6892 22.2701 100.00

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288 285

transient response is given in the 6-th column, and a BenchmarkSatisfaction Index (BSI) for the transient duration in the 7-thcolumn (100% means that the transient duration is less than 2 sand 0% corresponds to more than 4 s). The results show more orless good coherence between the simulation and experimentalresults. Certain discrepancy between the simulation and experi-mental results for the disturbance frequency of 50 Hz probablycomes from the modeling error of the secondary-path model,around this frequency, used in the simulator of the benchmark.

The simulation results for simple step test of Level 2 are given inTable 3 and the experimental results in Table 4. For Level 1 tests, apart

Table 3Simple step test (simulation)—Level 2.

Frequency (Hz) Global (dB) Dist. atte. (dB) Max. amp. (dB at Hz) N

50–70 35.5110 27.42–23.00 11.28 at 114.06 8755–75 38.6182 33.68–33.59 9.59 at 114.06 6760–80 39.8107 41.80–37.09 6.84 at 114.06 5365–85 39.9478 49.57–43.77 6.37 at 50.00 6470–90 38.4478 54.70–47.07 9.09 at 50.00 8075–95 35.1036 48.74–36.81 11.17 at 50.00 95

Table 4Simple step test (experimental results)—Level 2.

Frequency (Hz) Global (dB) Dist. atte. (dB) Max. amp. (dB at Hz) N

50–70 24.6660 20.58–17.49 18.06 at 131.25 1455–75 36.9297 34.20–30.54 18.88 at 129.69 1760–80 39.9376 44.32–37.43 18.00 at 134.37 665–85 32.5931 37.85–32.34 14.65 at 135.94 470–90 36.3403 55.54–47.05 14.41 at 137.50 575–95 33.7952 43.26–36.27 13.07 at 137.50 11

0 5 10

−0.05

0

0.05

Res

idua

l For

ce [V

]

Sim

0 5 10

−0.05

0

0.05

Res

idua

l For

ce [V

]

Step Frequen

0 5 10

−0.05

0

0.05

Tim

Res

idua

l For

ce [V

]

Ch

75 Hz

60 Hz 70 Hz 60 Hz

50 Hz

Fig. 7. Experimental time-domain res

from the disturbance at 50 Hz, disturbances at other frequencies arerejected. The global attenuation of more than 30 dB is metin simulation for all frequencies. However, in Level 2 real-timeexperiments the performance for the disturbance frequency pair(50–70) Hz is not good. The main reason is that the estimatedparameters in the adaptation algorithm do not converge to the truevalues (a linear controller with known disturbance frequencies per-forms very well in simulation as well as in real experiments). For theother disturbance frequencies the transient behavior in simulationand experiment are close and satisfy more or less the specifications.This effect can be reduced by adding a small damping to the internal

orm2 trans. (�10−3) Norm2 res. (�10−3) Max. val. (�10−3) BSI (%)

.6744 6.4925 61.5624 100.00

.8341 4.6086 59.1878 100.00

.7933 3.9936 53.7369 100.00

.8855 3.8930 68.9495 100.00

.7946 4.2437 76.4654 100.00

.3054 4.7593 72.0523 100.00

orm2 trans. (�10−3) Norm2 res. (�10−3) Max. val. (�10−3) BSI (%)

4.4955 34.0463 50.7286 100.004.1515 6.4665 86.2932 100.004.0941 4.0669 55.8595 100.007.4775 8.2762 54.6568 100.002.3746 4.7614 63.1648 100.006.2289 5.7348 86.3334 50.78

15 20 25 30

ple Step Test

Maximum value = 0.018578

Open LoopClosed Loop

15 20 25 30

cy Changes Test

Maximum value = 0.020664

15 20 25 30e [sec]

irp Test

Maximum value = 0.025549

50 Hz 60 Hz

95 Hz50 Hz

ponses for different Level 1 tests.

0 5 10 15 20 25 30

−0.05

0

0.05

Res

idua

l For

ce [V

]

Simple Step Test

Maximum value = 0.05586

Open LoopClosed Loop

0 5 10 15 20 25 30

−0.05

0

0.05

Res

idua

l For

ce [V

]

Step Changes in Frequency Test

Maximum value = 0.042226

0 5 10 15 20 25 30

−0.05

0

0.05

Time [sec]

Res

idua

l For

ce [V

]

Chirp Test

Maximum value = 0.050775

60−80 Hz

55−75 60−80 55−75 50−70 55−75

50−70 Hz75−95 Hz

50−70 Hz

Fig. 8. Experimental time-domain responses for different Level 2 tests.

0 50 100 150 200 250 300 350 400−100

−90

−80

−70

−60

−50

−40

Frequency [Hz]

PS

D e

stim

ate

[dB

]

Power Spectral Density Comparison

PSDOLPSDCL

Fig. 9. Power spectral density for the real-time simple step test with disturbancefrequency of 75 Hz (Level 1).

0 50 100 150 200 250 300 350 400−100

−90

−80

−70

−60

−50

−40

Frequency [Hz]

PS

D e

stim

ate

[dB

]

Power Spectral Density Comparison

PSDOLPSDCL

Fig. 10. Power spectral density for the real-time simple step test with disturbancefrequencies of 60 and 80 Hz (Level 2).

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288286

model of the disturbance. This way, a small error in the schedulingparameters will have less effect in the performance, but with the costof having less attenuation for the exact parameter estimates.

Although the maximum amplification of disturbance in simula-tion is close to that of linear controller, higher values are obtainedin real experiments. This probably comes from the modeling erroraround 129–137 Hz.

The real-time response from the simple step test with disturbancefrequency of 75 Hz is shown in the first plot of Fig. 7. Similarly, inFig. 8 the first plot presents the simple step test response for Level2 with disturbance frequencies of 60 and 80 Hz. Fig. 9 illustrates thecomparison between the open-loop (dashed) and the closed-loop

power spectral density for the real-time simple step test with singledisturbance frequency of 75 Hz. Strong attenuation (around 45 dB) at75 Hz and low (or no) amplification at other frequencies can beobserved. Similar conclusion can be drawn from Fig. 10 for the simplestep test of Level 2 with disturbance frequencies of 60 and 80 Hz.

4.2. Step changes in frequencies test

For Level 1 of the benchmark, three sequences of step changes inthe frequency of the disturbance are considered. These sequences

Table 5Step changes in frequencies test (simulation).

Frequency (Hz) Norm2 trans. (�10−3) Max. val. (�10−3)

Level 1Sequence 1

60-70 7.9407 20.756570-60 8.3077 16.510960-50 10.2125 13.490350-60 7.3136 15.6196

Sequence 275-85 7.2684 13.934385-75 8.7004 17.002575-65 7.6486 17.049165-75 7.8875 15.2766

Sequence 385-95 8.3668 20.256795-85 9.5780 18.537685-75 7.3373 17.121875-85 8.0514 17.4093

Level 2Sequence 1

½55–75�-½60–80� 10.0628 22.8093½60–80�-½55–75� 16.3247 21.5839½55–75�-½50–70� 20.0613 18.7106½50–70�-½55–75� 10.0798 17.8500

Sequence 2½70–90�-½75–95� 11.5762 18.1048½75–95�-½70–90� 13.9285 22.5232½70–90�-½65–85� 12.0298 23.1736½65–85�-½70–90� 10.8733 20.8162

Table 6Step changes in frequencies test (experimental results).

Frequency (Hz) Norm2 trans. (�10−3) Max. val. (�10−3)

Level 1Sequence 1

60-70 9.8243 19.728970-60 9.8454 18.244560-50 22.6091 18.221050-60 13.8104 19.4511

Sequence 275-85 8.5829 15.783385-75 10.1036 17.321875-65 9.9323 19.739165-75 10.1643 19.72531

Sequence 385-95 8.5971 15.835995-85 9.8947 17.252385-75 9.1527 16.069875-85 9.1745 17.0129

Level 2Sequence 1

½55–75�-½60–80� 13.7017 22.5980½60–80�-½55–75� 18.0644 23.9297½55–75�-½50–70� 51.0763 26.2669½50–70�-½55–75� 14.2444 20.1445

Sequence 2½70–90�-½75–95� 13.0523 18.9047½75–95�-½70–90� 12.7703 21.6059½70–90�-½65–85� 15.5371 20.3018½65–85�-½70–90� 13.3272 18.9160

Table 7Chirp changes.

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288 287

are defined as follows:

Error

Sequence 1: 60-70-60-50-60

−3 −6

Sequence 2: 75-85-75-65-75

Maximum value (�10 ) Mean square value (�10 )

Sequence 3: 85-95-85-75-85

Level 1—sim 6.40 3.5910Level 1—exp 7.54 4.5412Level 2—sim 10.12 10.5170Level 2—exp 11.56 11.8759

Similarly, for Level 2, two sequences of the step changes in thedisturbance frequencies are defined (see the first column of Table 5).The transient performance in simulation for Levels 1 and 2 are givenin Table 5 and the experimental results in Table 6. It can be observedthat good performance is obtained for all disturbance frequency pairsexcept for (50–70) in the real experiment.

The second plot of Figs. 7 and 8 present the real-time responsefor the first disturbance frequency sequence of the step changes infrequencies test for Levels 1 and 2, respectively.

4.3. Chirp test

For Level 1 of the benchmark a chirp signal that starts from50 Hz and goes to 95 Hz and returns to 50 Hz with a variation rateof 10 Hz/s is applied as the disturbance signal. For Level 2 thedisturbance frequencies change from 50–70 Hz to 75–95 Hz with avariation rate of 5 Hz/sec and return to 50–70 Hz. The maximumvalue and the two-norm of the disturbance response in simulationand in the real-time experiment are given in Table 7. Theexperimental results of the chirp disturbance responses for Levels1 and 2 are given in Figs. 7 and 8, respectively.

5. Conclusions

A new method for fixed-order gain-scheduled H∞ controllerdesign is proposed and applied to the active suspension benchmark.It is shown that one or two unknown sinusoidal disturbances can be

rejected using the gain-scheduled controller and an adaptationalgorithm that estimates the internal model of the disturbance. Theproposed gain-scheduled controller design method is able to satisfyall frequency-domain constraints. However, the results are slightlydeteriorated in simulation and real experiments. The main reasonsare the followings:

�

During the convergence of the scheduling parameter, the wholesystem becomes nonlinear and the desired performance is notachieved.

�

Even at the steady state, there is always an estimation error inthe scheduling parameter.

�

The modeling error in the secondary-path model is not con-sidered in the design.

Although the proposed method could consider the modeling errorin the design, it has not been taken to account for some reasons.First, it was supposed that the provided model for the benchmarkis very close to the real system and modeling error can beneglected. Second, considering the unmodelled dynamics makesthe optimization method more complicated (number of con-straints increases) and, finally, robust controllers lead generallyto conservative solutions to the detriment of performance.

A. Karimi, Z. Emedi / European Journal of Control 19 (2013) 279–288288

Acknowledgments

The authors would like to thank Prof. Landau for his advicesand good discussions and Abraham Castellanos Silva for his helpand support during the real-time experiments.

References

[1] P. Apkarian, P. Gahinet, G. Becker, Self-scheduled H∞ control of linearparameter-varying systems: a design example, Automatica 31 (1995)1251–1261.

[2] P. Arcara, S. Bittanti, M. Lovera, Periodic control of helicopter rotors forattenuation of vibrations in forward flight, IEEE Transactions on ControlSystems Technology 8 (2000) 883–894.

[3] P. Ballesteros, X. Shu, W. Heins, C. Bohn, LPV gain-scheduled output feedbackfor active control of harmonic disturbances with time-varying frequencies, in:Advances on Analysis and Control of Vibrations—Theory and Applications,Intechopen.com, Rijeka, Croatia, 2012.

[4] M. Bodson, A. Sacks, P. Khosla, Harmonic generation in adaptive feedforwardcancelation schemes, IEEE Transactions on Automatic Control 33 (1994)2213–2221.

[5] M. Bodson, Rejection of periodic disturbances of unknown and time-varyingfrequency, International Journal of Adaptive Control and Signal Processing 19(2005) 67–88.

[6] G. Calafiore, M.C. Campi, The scenario approach to robust control design, IEEETransactions on Automatic Control 51 (2006) 742–753.

[7] K.K. Chew, M. Tomizuka, Digital control of repetitive errors in disk drivesystems, in: American Control Conference, Pittsburgh, PA, USA, pp. 540–548.

[8] C.J. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, MacMillan,New York, 1992.

[9] H. Du, L. Zhang, Z. Lu, X. Shi, LPV technique for the rejection of sinusoidaldisturbance with time-varying frequency, IEE Proceedings—Control Theoryand Applications 150 (2003) 132–138.

[10] Z. Emedi, A. Karimi, Fixed-order LPV controller design for rejection of asinusoidal disturbance with time-varying frequency, in: IEEE Multi-Conference on Systems and Control, Dubrovnik, Croatia.

[11] P. Gahinet, P. Apkarian, A linear matrix inequality approach to H∞ control,International Journal of Robust and Nonlinear Control 4 (1994) 421–448.

[12] G. Galdos, A. Karimi, R. Longchamp, Robust controller design by convexoptimization based on finite frequency samples of spectral models, in: 49thIEEE Conference on Decision and Control, Atlanta, USA.

[13] W. Heins, P. Ballesteros, X. Shu, C. Bohn, LPV gain-scheduled observer-basedstate feedback for active control of harmonic disturbances with time-varyingfrequencies, in: Advances on Analysis and Control of Vibrations—Theory andApplications, Intechopen.com, Rijeka, Croatia, 2012.

[14] A. Karimi, G. Galdos, Fixed-order H∞ controller design for nonparametricmodels by convex optimization, Automatica 46 (2010) 1388–1394.

[15] A. Karimi, Frequency-domain robust controller design: a toolbox for MATLAB,2012. URL: ⟨http://la.epfl.ch/FDRC_Toolbox⟩.

[16] S.M. Kuo, D.R. Morgan, Active Noise Control Systems: Algorithms and DSPImplementations, John Wiley & Sons, Inc., New York, NY, USA, 1995.

[17] I.D. Landau, A. Constantinescu, D. Rey, Adaptive narrow band disturbancerejection applied to an active suspension—an internal model principleapproach, Automatica 41 (2005) 563–574.

[18] I.D. Landau, R. Lozano, M. M’Saad, A. Karimi, Adaptive Control: Algorithms,Analysis and Applications, Springer-Verlag, London, 2011.

[19] I.D. Landau, T.B. Airimitoaie, A.C. Silva, Gabriel Buche, Benchmark on AdaptiveRegulation – Rejection of unknown/time-varying multiple narrow banddisturbances, in: European Control Conference, Zurich, Switzerland, submittedfor publication. URL: ⟨http://www.gipsa-lab.grenoble-inp.fr/� ioandore.landau/benchmark_adaptive_regulation⟩.

[20] P. Mäkilä, Approximation of stable systems by laguerre filters, Automatica 26(1990) 333–345.

[21] A. Sacks, M. Bodson, P. Khosla, Experimental results of adaptive periodicdisturbance cancelation in a high performance magnetic disk drive, ASMEJournal of Dynamic Systems Measurement and Control 118 (1996) 416–424.