Automatica 46 (2010) 1388–1394

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Automatica

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Brief paper

Fixed-order H∞ controller design for nonparametric models by convexoptimizationI

Alireza Karimi ∗, Gorka GaldosLaboratoire d’Automatique, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

a r t i c l e i n f o

Article history:Received 18 May 2009Received in revised form19 April 2010Accepted 28 April 2010Available online 8 June 2010

Keywords:H∞ controlData-based controlConvex optimization

a b s t r a c t

A new approach for robust fixed-order H∞ controller design by convex optimization is proposed. Lineartime-invariant single-input single-output systems represented bynonparametricmodels in the frequencydomain are considered. It is shown that theH∞ robust performance condition can be represented by a setof linear or convex constraints with respect to the parameters of a linearly parameterized controller inthe Nyquist diagram. Systems with time-delay and with multimodel and frequency-domain uncertaintycan be considered straightforwardly in the proposed approach. The proposed method is compared withthe standard H∞ control problem by a simulation example on an unstable system.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Most controller design methods are based on plant parametricmodels. A parametric model can be obtained either by first prin-ciple modeling or by parameter estimation techniques using mea-sured data. However, it is usually too difficult or time consumingto obtain a parametric model based on physical laws. On the otherhand, identification of parametric models is based on several a pri-ori information and user choices like sampling period, time-delay,number of parameters in numerator and denominator of plant andnoise model, optimal excitation etc. For these reasons, some data-based methods (in time-domain or in frequency-domain) for con-troller design have been developed. Iterative Feedback Tuning (IFT)(Hjalmarsson, Gevers, Gunnarsson, & Lequin, 1998), Virtual Refer-ence Feedback Tuning (VRFT) (Campi, Lecchini, & Savaresi, 2002)and Iterative Correlation-based Controller Tuning (ICbT) (Karimi,Mišković, & Bonvin, 2004) use time-domain data to tune fixed-order controllers for model reference and tracking problem. LQGcontrol and optimal tracking in data space are considered in Skel-ton and Shi (1994) and Yasumasa, Duanm, and Ikeda (2005), re-spectively. A method for stability analysis of linear discrete time

I This research work is financially supported by the Swiss National ScienceFoundation under Grant No. 200020-107872. The material in this paper waspartially presented at the 47th IEEE Conference on Decision and Control, Cancun,Mexico, December 9–11, 2008. This paper was recommended for publication inrevised form by Associate Editor Tong Zhou under the direction of Editor RobertoTempo.∗ Corresponding author. Tel.: +41 21 693 3841; fax: +41 21 693 2574.E-mail address: Alireza.Karimi@epfl.ch (A. Karimi).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.05.019

systems using only time-domain data is proposed in Park and Ikeda(2004).Frequency-domain data or spectral models can be easily ob-

tained from input/output data using Fourier or spectral analysis(Pintelon & Schoukens, 2001). In this type of models the informa-tion is not condensed into a small set of parameters thus avoidingerrors of unmodelled dynamics that appear in parametric models.Moreover, an estimate of the uncertainty due to noise can be read-ily computed.Although spectralmodels are largely used in practice, controller

design methods based on this type of models are rather limited.The first systematic controller design methods were based on loopshaping with graphical tools in Bode diagrams or in Nichols chartand are discussed in classical textbooks for design and analysis ofcontrol systems. These approaches are very intuitive andworkwellfor simple systems that can be approximated by a low-ordermodelwith relatively small delay. For unstable and nonminimum-phasesystems and systems with parametric and frequency-domain un-certainty, more advancedmethods should be used. Recently, it hasbeen shown that the set of all stabilizing PID controllers achievinga desired gain and phase margin or H∞ norm can be obtained us-ing only the frequency-domain data (Keel & Bhattacharyya, 2008).Another frequency-domain method is the well-known Quantita-tive Feedback Theory (QFT) (Horowitz, 1993) which is based onloop shaping in the Nichols chart. Frequency-domain approacheslead usually to low-order controllers and the design proceduresneed some expertise and are based on trial and error. Although re-cently optimization approaches are used to compute controllers inthe QFT framework (Bryant & Halikias, 1995; Chait, Chen, & Hollot,1999; Halikias, Zolotas, & Nandakumar, 2007), H2 and H∞ controlcriteria for spectral models have not been considered.

A. Karimi, G. Galdos / Automatica 46 (2010) 1388–1394 1389

With new progress in numerical methods for solving convexoptimization problems, new approaches for controller design withconvex objectives and constraints have been developed. Thesetechniques have been also applied to controller design for spectralmodels. In Grassi and Tsakalis (1996) and Grassi et al. (2001) aconvex optimization method for PID controller tuning by open-loop shaping in the frequency-domain is proposed. The infinity-norm of the difference between the desired open-loop transferfunction and the achieved one weighted by a so-called targetsensitivity function is minimized. For open-loop stable systems, itis shown through the small gain theorem that if the infinity norm isless than 1, then the nominal closed-loop system is stable. This isa sufficient condition which depends on the choice of the targetsensitivity function. The condition for the stability of multiplemodels becomesmore conservative as for eachmodel a reasonabletarget sensitivity function should be available.In Karimi, Kunze, and Longchamp (2007) a robust fixed-order

controller design using linear programming is proposed. The mainfeature of this method is that the stability and some robustnessmargins are guaranteed by linear constraints in the Nyquist di-agram and the method is applicable to multiple models as well.However, the performance specifications are limited to the choiceof a lower bound for crossover frequency and minimization of theintegral of the tracking error. The results are improved by open-loop and closed-loop shaping using quadratic programming inGaldos, Karimi, and Longchamp (2007).In this paper, a new approach for robust fixed-order controller

design is developed. It is shown that robust fixed-order linearlyparameterized controllers for Linear Time Invariant Single-InputSingle Output (LTI-SISO) systems represented by nonparametricspectral models can be computed by convex optimization. Theperformance specification, like the standard H∞ control problem,is a constraint on the infinity norm of the weighted sensitivityfunction. It should be mentioned that the set of all fixed-orderstabilizing controllers is a nonconvex set. In this paper, an innerconvex approximation of this set is given by a set of linear con-straints in the Nyquist diagram. The proposed method can be usedfor PID controllers aswell as for higher order linearly parametrizedcontrollers in discrete or continuous time. The case of unstableopen-loop systems can also be considered if a stabilizing controlleris available or the number of unstable poles of the plant is known.The main idea is to define new constraints such that the designedopen-loop system has the winding number satisfying the Nyquiststability criterion. Another important feature is that, by contrastwith the standard H∞ problem, this approach can treat the case ofmultimodel uncertainty and systems with time-delay.This paper is organized as follows: In Section 2 the class ofmod-

els, controllers and the control objectives are defined. Section 3 in-troduces the control design methodology based on the linear andconvex constraints in the Nyquist diagram. Simulation results andcomparisonwith the standardH∞ design are given in Section 4 andconcluding remarks in Section 5.

2. Problem formulation

2.1. Class of models

The class of causal continuous-time LTI-SISO systems withbounded infinity norm is considered. It is assumed that the plantmodel belongs to a setG that containsm spectralmodelswithmul-tiplicative unstructured uncertainty:

G = {Gi(jω)[1+W2i(jω)∆]; i = 1, . . . ,m;ω ∈ R} (1)

where W2i(jω) is the uncertainty weighting frequency functionand ∆ is a stable transfer function with ‖∆‖∞ < 1. This type ofmodels can be obtained from a parametric model or by spectralanalysis from a set of input/output data.

In the sequel, for the sake of simplicity, we consider one of themodels in G with multiplicative frequency-domain uncertainty,G(jω)[1 + W2(jω)∆] and a continuous-time controller will bedesigned. Then the results are extended to themultimodel case andthe convex combination of m spectral models. The results are alsoapplicable to discrete-time models and other type of frequency-domain uncertainty.

2.2. Class of controllers

Linearly parameterized controllers are given by:

K(s, ρ) = ρTφ(s) (2)whereρT = [ρ1, ρ2, . . . , ρn],

φT (s) = [φ0(s), φ1(s), . . . , φn−1(s)],n is the number of controller parameters and φi(s) are stable trans-fer functions with bounded infinity norm. The optimal choice ofφi(s) is highly related to the desired performance and the plantmodel. However, they may be chosen from a set of generalized or-thonormal basis functions. Consider for example the Laguerre ba-sis Akcay and Ninness (1999) and Mäkilä (1990):

φ0(s) = 1, φi(s) =√2ξ(s− ξ)i−1

(s+ ξ)ifor i ≥ 1 (3)

with ξ > 0. It can be shown that for any stable rational finite or-der transfer function F(s) and for arbitrary ε > 0 there exists asufficiently large n such that‖F − ρTφ‖p < ε for 0 < p <∞.Therefore, with this controller parameterization, any finite orderstable transfer function can be approximated with a desired accu-racy by increasing the number of controller parameters. The qual-ity of this approximation for a finite n, however, depends on thedifference between the poles of F(s) and ξ . An appropriate choiceof ξ can lead to a better approximation for a given controller order.The optimal choice of basis functions has already been investigatedin the context of modeling and identification (Heuberger, Van DenHof, &Wahlberg, 2004) andwill not be considered in this contribu-tion. However, a practical guideline for an appropriate choice of thebasis functions for an n-th order controller (with n+1 parameters)follows:(1) Choose Laguerre basis with a large dimension, nmax+1, and anarbitrary value for ξ > 0.

(2) Compute a nmax-th order controller by the proposed optimiza-tion problem.

(3) Determine n dominant poles, ξ1, . . . , ξn, of the nmax-ordercontroller (using the model reduction techniques).

(4) Use the following generalized orthonormal basis functions fori = 1, . . . , n (Akcay & Ninness, 1999):

φ0(s) = 1, φi(s) =√2Re{ξi}s+ ξi

i−1∏k=1

s− ξ̄ks+ ξk

where ξ̄k is the complex conjugate of ξk.The main reason to use a linearly parameterized controller in

this paper is that every point on the Nyquist diagram of the open-loop transfer function L(jω, ρ) can be written as a linear functionof the controller parameters ρ:

L(jω, ρ) = K(jω, ρ)G(jω) = ρTφ(jω)G(jω). (4)This property helps obtaining a convex parameterization of fixed-order H∞ controllers.

Remark. The bounded infinity-norm condition will be relaxed toallow the possible poles on the imaginary axis for plant model andcontroller. It is clear that, in this case, PID controllers belong to theset of parameterized controllers.

1390 A. Karimi, G. Galdos / Automatica 46 (2010) 1388–1394

2.3. Design specifications

Let the sensitivity function S(s) = [1 + L(s)]−1 and the com-plementary sensitivity functionT (s) = L(s)[1+L(s)]−1 be defined.A very standard robust control problem is to design a controllerthat satisfies ‖W1S‖∞ < 1 for a set of models, whereW1(s) is theperformance weighting filter. If the set of models is represented bymultiplicative uncertainty, the necessary and sufficient conditionfor robust performance is given byDoyle, Francis, and Tannenbaum(1992):‖|W1S| + |W2T |‖∞ < 1. (5)There is no analytical solution to this problem, however, in thestandard H∞ framework a solution to the following approximateproblem can be found:∥∥∥∥W1SW2T

∥∥∥∥∞

<1√2. (6)

This solution is conservative and leads to high order controllers.Moreover, it cannot be applied to systems with multimodel uncer-tainty.The proposed approach in this paper is based on an infinite

number of linear or convex constraints on the Nyquist diagramsuch that the following robust performance constraint is satisfied.|W1(jω)S(jω)| + |W2(jω)T (jω)| < 1 ∀ω. (7)

3. Robust controller design in Nyquist diagram

3.1. Robust performance constraints

The basic idea is to represent the robust performance con-straints in (7) in the Nyquist diagram and give a set of linear orconvex constraints which guarantee that the robust performancecondition is satisfied. Thisway, the controller design is representedby a convex feasibility problem.Multiplying the robust performance condition in (7) by |1 +

L(jω, ρ)| gives:|W1(jω)| + |W2(jω)L(jω, ρ)| < |1+ L(jω, ρ)| ∀ω. (8)Note that |1 + L(jω, ρ)| is the distance between the critical pointand L(jω, ρ). Hence, this constraint is satisfied if and only if thereis no intersection in the Nyquist diagram between a circle centeredat the critical point with a radius of |W1(jω)| and a circle centeredat L(jω, ρ) with a radius of |W2(jω)L(jω, ρ)| for all ω (Doyle et al.,1992).Now, consider a straight line d∗(ω)which is tangent to the circle

with radius |W1(jω)| and orthogonal to the line between the criti-cal point and L(jω, ρ). Therefore, the robust performance conditionin (7) is satisfied if and only if the circle centered at L(jω, ρ) doesnot intersect d∗(ω) and is completely in the side that excludes thecritical point (at the right hand side in Fig. 1). This condition cannotbe represented as a convex constraint because d∗(ω) is a functionof the controller parameters. Suppose that the frequency responseof a desired open-loop transfer function, Ld(jω), is available. Then,d∗(ω) can be approximated by d(ω) which is tangent to the circlewith radius |W1(jω)| but orthogonal to the line connecting the crit-ical point to Ld(jω) (see Fig. 1). This will be a good approximationif Ld(jω) is ‘‘close’’ to L(jω, ρ).It should be noted that the equation of d(ω) at each frequency

depends only on W1(jω) and Ld(jω). If we name x and y, respec-tively, the real and imaginary parts of a point on the complex plane,the equation of d(ω) at each frequency becomes:|W1(jω)[1+ Ld(jω)]| − Im{Ld(jω)}y− [1+ Re{Ld(jω)}][1+ x] = 0 (9)

where Re{·} and Im{·} represent real and imaginary parts of a com-plex value, respectively. Therefore, the condition that L(jω, ρ) forall ω is located in the side of d(ω) that excludes the critical pointcan be given by the following linear constraints:

Fig. 1. Linear constraints for robust performance in the Nyquist diagram.

|W1(jω)[1+ Ld(jω)]| − Im{Ld(jω)}Im{L(jω, ρ)}− [1+ Re{Ld(jω)}][1+ Re{L(jω, ρ)}] < 0 ∀ω.

Replacing Re{Ld(jω)} = 1/2[Ld(jω) + Ld(−jω)] and a similar ex-pression for the imaginary part, the above linear constraints canbe further simplified to:

|W1(jω)[1+ Ld(jω)]|− Re{[1+ Ld(−jω)][1+ L(jω, ρ)]} < 0 ∀ω. (10)

There exist two alternatives in order that this condition to besatisfied for allmodels in the uncertainty set represented by a circlecentered at L(jω, ρ). The first alternative is to approximate theuncertainty circle by a polygon of q > 2 vertices. Then, the robustperformance condition in (7) is satisfied if all vertices are locatedin the right side of d(ω). This can be represented by the followinglinear constraints:

|W1(jω)[1+Ld(jω)]|−Re{[1+ Ld(−jω)][1+Li(jω, ρ)]} < 0∀ω and i = 1, . . . , q (11)

where Li(jω, ρ) = K(jω, ρ)Gi(jω) and

Gi(jω) = G(jω)[1+|W2(jω)|cos(π/q)

ej2π i/q]. (12)

It can be observed that the number of linear constraints are multi-plied by q when the uncertainty circle is approximated by a poly-gon of q vertices.The second alternative is to increase the radius of the perfor-

mance circle by |W2(jω)L(jω, ρ)|which leads to the following con-vex constraints:

|W1(jω)[1+ Ld(jω)]| + |W2(jω)L(jω, ρ)[1+ Ld(jω)]|− Re{[1+ Ld(−jω)][1+ L(jω, ρ)]} < 0 ∀ω. (13)

This alternative has less constraints and no conservatism but leadsto a slightly more complex convex optimization problem (convexconstraints instead of linear constraints).The nonconvex constraint in (5) is convexified using a desired

open-loop transfer function Ld(s). In other words, the convex setin (13) is an inner approximation of the nonconvex set definedby the constraint in (5). The following proposition shows underwhich condition a feasible point of the nonconvex set in (5) is alsoa feasible point of the convex inner approximation set in (13).

Proposition 1. Consider that ρ◦ belongs to the non-convex set (5),i.e.:

‖ |W1S(ρ◦)| + |W2T (ρ◦)| ‖∞ = γ (ρ◦) < 1 (14)

A. Karimi, G. Galdos / Automatica 46 (2010) 1388–1394 1391

Fig. 2. A graphical illustration of Proposition 1 for γ (ρ◦) = 0.7.

then ρ◦ satisfies the constraints in (13) if and only if:

|6 (1+ Ld(jω))− 6 (1+ L(jω, ρ◦))|

< cos−1 (|W1(jω)S(jω, ρ◦)| + |W2(jω)T (jω, ρ◦)|) ∀ω. (15)

The above inequality is satisfied if

|6 [1+ L(jω, ρ◦)] − 6 [1+ Ld(jω)]| < cos−1 γ (ρ◦) ∀ω. (16)

Proof. The proof is straightforward using the following relation:

Re{[1+ Ld(−jω)][1+ L(jω, ρ◦)]}

= |1+ Ld(−jω)||1+ L(jω, ρ◦)| cosα (17)

where

α = |6 [1+ L(jω, ρ◦)] − 6 [1+ Ld(jω)]|. (18)

Replacing the right hand side of (17) in (13) gives:

|W1(jω)| + |W2(jω)L(jω, ρ◦)| < |1+ L(jω, ρ◦)| cosα ∀ω. (19)

Dividing the both sides by |1+ L(jω, ρ◦)| leads to:

|W1(jω)S(jω, ρ◦)| + |W2(jω)T (jω, ρ◦)| < cosα ∀ω

which is equivalent to (15). A sufficient condition for the above in-equality is that themaximum value of the left hand side be smallerthan cosα or:

γ (ρ◦) < cosα

from which (16) can be concluded. �

Suppose for example thatρ◦ is a feasible point of the nonconvexset with γ (ρ◦) = 0.7, then α, the phase difference of 1 + Ld(jω)and 1 + L(jω, ρ◦), should be less than cos−1 0.7 = 45◦. Thisrepresents a very large set (one quarter of the complex plane)of admissible Ld(jω) for which ρ◦ is in the feasibility set of theinner approximation (see Fig. 2). It is clear that if the specificationsare too tight so that for any feasible point ρ◦, γ (ρ◦) is very closeto 1, the non convex set in (5) is too small and finding an innerapproximation by the choice of Ld becomes very difficult. However,milder specifications (e.g. by reducing the gain ofW1 andW2) leadsto a larger nonconvex set in (5) and a reasonable choice of Ld leadsusually to a nonempty inner approximation of the nonconvex set.

3.2. Main result

The main result of this section is presented in the followingtheorem:

Theorem 1. Given the set of models G in (1) with performanceweighting functions W1i(jω), the linearly parameterized controllerin (2) stabilizes all models in G and satisfies the following robustperformance condition:

‖ |W1iSi| + |W2iTi| ‖∞ < 1 for i = 1, . . . ,m (20)

if

|W1i(jω)| + |W2i(jω)ρTφ(jω)Gi(jω)|

−Re{[1+ Ldi(−jω)][1+ ρ

Tφ(jω)Gi(jω)]}|1+ Ldi(jω)|

< 0

∀ω for i = 1, . . . ,m (21)

where Ldi(jω) is chosen such that the number of counterclockwiseencirclements of the critical point by its Nyquist plot is equal to thenumber of unstable poles of Gi(s).Proof. Since the real value of a complex number is less than orequal to its magnitude, we have:

Re{[1+ Ldi(−jω)][1+ ρTφ(jω)Gi(jω)]}

≤ |[1+ Ldi(−jω)][1+ ρTφ(jω)Gi(jω)]|. (22)

Then from (21) we obtain:

|W1i(jω)| + |W2i(jω)ρTφ(jω)Gi(jω)|

− |1+ ρTφ(jω)Gi(jω)| < 0 ∀ω for i = 1, . . . ,m (23)

which gives:∣∣∣∣ W1i(jω)1+ Li(jω, ρ)

∣∣∣∣+ ∣∣∣∣W2i(jω)Li(jω, ρ)1+ Li(jω, ρ)

∣∣∣∣ < 1∀ω for i = 1, . . . ,m (24)

that leads directly to (20).Now we should show that this controller stabilizes all models

in G. From (21), for i = 1, . . . ,m, we have:

Re{[1+ Ldi(−jω)][1+ ρTφ(jω)Gi(jω)]} > 0 ∀ω (25)

orwno{[1+ Ldi(−jω)][1+ Li(jω, ρ)]

}= 0,wherewno stands for

winding number around the origin. It should be mentioned thatLdi(−jω) and Li(jω, ρ) are zero or constant for the semicircle withinfinity radius of the Nyquist contour so the wno depends only onthe variation of s on the imaginary axis. Therefore:

wno[1+ Ldi(jω)] = wno[1+ Li(jω, ρ)]. (26)

Since Ldi(jω) satisfies the Nyquist criterion, Li(jω, ρ) will do so aswell and all closed-loop systems are stable. �

Corollary 1. Consider the convex combination of m spectral modelsin G:m∑i=1

λiGi(jω)[1+W2i(jω)∆] , Gλ(jω)[1+W2(jω)∆]

where

Gλ(jω) ,m∑i=1

λiGi(jω)

W2(jω) ,

m∑i=1λiGi(jω)W2i(jω)

Gλ(jω)

λ = [λ1, . . . , λm],∑mi=1 λi = 1 and λi ∈ [0, 1]. Then, the linearly

parameterized controller in (2)will stabilize this model for any admis-sible λ and satisfies the following robust performance condition:

‖ |W1S| + |W2T | ‖∞ < 1 (27)

where

1392 A. Karimi, G. Galdos / Automatica 46 (2010) 1388–1394

W1(jω) ,m∑i=1

λiW1i(jω)

if (21) is satisfied with Ldi(jω) = Ld(jω) for i = 1, . . . ,m. Ld(jω)should be chosen such that the number of counterclockwise encir-clements of the critical point by its Nyquist plot is equal to the numberof unstable poles of Gλ(s). A fixed Ld(jω) means that the number ofunstable poles of Gλ(s) should be fixed for all λ.

Proof. Multiplying (21) by λi and adding the m constraints weobtain:m∑i=1

λi|W1i(jω)| +m∑i=1

|W2i(jω)ρTφ(jω)λiGi(jω)|

−

Re

{[1+ Ld(−jω)]

[1+ ρTφ(jω)

m∑i=1λiGi(jω)

]}|1+ Ld(jω)|

< 0

∀ω. (28)

We have: |W1(jω)| ≤∑mi=1 λi|W1i(jω)| and∣∣∣∣∣ρTφ(jω) m∑

i=1

λiGi(jω)W2i(jω)

∣∣∣∣∣≤

m∑i=1

|W2i(jω)ρTφ(jω)λiGi(jω)|. (29)

Therefore:|W1(jω)| +

∣∣ρTφ(jω)Gλ(jω)W2(jω)∣∣−Re{[1+ Ld(−jω)][1+ ρTφ(jω)Gλ(jω)]}

|1+ Ld(jω)|< 0 ∀ω. (30)

The rest of the proof is similar to that of Theorem 1. �

Remarks.(1) The results of Theorem 1 are valid if Li(s, ρ) has some poleson the imaginary axis, say {jp1, jp2, . . .}. In this case ω ∈ R −{[p1−ε, p1+ε], [p2−ε, p2+ε], . . .}where ε is a small positivevalue. The stability is guaranteed if Ldi(s) contains the poleson the imaginary axis of Li(s, ρ) because they will have thesame behavior at the small semicircular detour of the Nyquistcontour at these poles.

(2) The same approach can be applied while an additive uncer-tainty model is available i.e.

G̃i(s) = Gi(s)+W3i(s)∆(s).

The robust performance condition is given by:∥∥∥∥ |W1iSi| + ∣∣∣∣W3iGi Ti

∣∣∣∣∥∥∥∥∞

< 1 for i = 1, . . . ,m. (31)

In this case the convex constraints in (21) can be used with thedifference that|W2i(jω)| = |W3i(jω)|/|Gi(jω)|.

(3) Individual shaping of the sensitivity functions is also possibleusing the constraints in (21) with one of the filters equal tozero.

(4) The robust performance can be improved by minimizing theupper bound of the infinity norm of the weighted sensitivityfunction. Consider following optimization problem for a singlemodel:min γ‖ |W1S| + |W2T | ‖∞ < γ . (32)

This optimization can be solved by an iterative bisection algo-rithm. At each iteration for a fixed γi, we replace W1 and W2

with W1/γi and W2/γi and we solve the feasibility problemrepresented by the linear constraints in (11) or convex con-straints in (13). If the problem is feasible γi+1 will be chosensmaller than γi and if the problem is infeasible γi+1 will beincreased.

3.3. How to deal with infinite number of constraints

It is shown in Theorem 1 that the problem of robust controllerdesign for systems with multimodel and frequency-domainuncertainty can be formulated as a convex feasibility problem (orlinear feasibility problem if we approximate the uncertainty circleby a polygon) with an infinite number of constraints. This problemis known as a convex (or linear) semi-infinite program (SIP) forwhich different numerical solutions exist in the literature (seeGoberna and Lopez (2002) for a survey).A practical solution is to choose a finite number of frequencies

and find a feasible solution for the constraints in (21) for ω ∈{ω1, ω2, . . . , ωN}. It is clear thatN should be sufficiently large suchthat the Nyquist diagram of L(jωk, ρ) is a good approximation ofL(jω, ρ).If the spectral models are obtained from a set of noisy data,

then the frequency-domain uncertainty sets are defined with aprobability level. In this case, even a feasible solution to the semiinfinite program will guarantee the robust performance with aprobability level. Therefore, it is more reasonable to use a random-ized approach to solve the SIP. According to the results of Calafioreand Campi (2006) and Campi and Garatti (2008) with a reasonablenumber N of randomly chosen frequency samples, the optimal so-lution ρ∗ to the convex optimization problem will satisfy the con-straints for all frequencies with a high probability level. In orderto be more precise, let the violation probability V (ρ∗) be definedas the probability that for ω0 ∈ R the convex constraints are notsatisfied for ρ∗. Then it can be shown that:

P {V (ρ∗) > ε} ≤

n−1∑i=0

(Ni

)ε i(1− ε)N−i (33)

whereP {·} stands for the probability of an event and ε is a satisfy-ing level. Consider, for example, PID controller design (n = 3) withN = 500 frequency points. Then, having a violation probability ofgreater than ε = 0.01 has a probability of less than 0.1234. Thisupper bound goes exponentially to zero with N . Therefore, the up-per bound can be reduced to 0.0027 forN = 1000 and to 4.2×10−7for N = 2000.

3.4. Choice of Ld(s)

It was shown in Proposition 1 that if the specifications are nottoo tight, for a large set of admissible Ld(s) a nonempty innerconvex approximation of the nonconvex set can be obtained if Ld(s)is ‘‘close’’ to L(s, ρ). Suppose that ρ∗ is the optimal solution of thenonconvex problem. It is well known that the optimal H∞ solutionis based on cancellation of stable poles and zeros of the plantby the controller. Therefore, an Ld(s) that contains the unstablepoles and zeros of of the plant model and controller (includingthe poles on the imaginary axis) will be ‘‘close’’ to L(s, ρ∗) andthe convex set generated based on this Ld(s)will likely contain theoptimal controller. In this case the optimal controller can be foundby minimizing a criterion J(ρ) = ‖L(ρ) − Ld‖ under the robustperformance constraint in (21). Since L(ρ) is linear with respect toρ, any norm of L(ρ) − Ld is a convex function of ρ. For example,if we design a PID controller for open-loop stable systems with nopole on the imaginary axis a good choice is Ld(s) = ωc/s with ωcthe desired closed-loop crossover frequency.If the first choice of Ld(jω) leads to a non feasible set, the iter-

ative windsurfing approach (Anderson, 2002) can be used to com-pute an appropriate Ld(s). In this approach we start with modestspecifications by reducing the gain ofW1 andW2 so that a feasiblesolution ρ1 is obtained. Then Ld(jω) = L(jω, ρ1) is chosen and the

A. Karimi, G. Galdos / Automatica 46 (2010) 1388–1394 1393

specifications will be tightened by increasing the gain of W1 andW2. A feasible solution ρ2 for the second feasibility problem willbe used to compute a new Ld(jω) = L(jω, ρ2). Although the con-vergence of this iterative approach to the optimal solution cannotbe proved, good results in practice can be obtained.The choice of Ld(s) is more important for unstable systems.

In this case, according to Theorem 1, the winding number of theNyquist plot of Ld(s) around the critical point should satisfy theNyquist stability criterion. For this purpose, the number of unstablepoles of the plantmodel should be knownor a stabilizing controllerK0(s) should be available. In the latter, Ld(s) = K0(s)G(s) is a goodchoice that satisfies the Nyquist criterion.

4. Simulation results

This example is taken from Djaferis (1995) where a robustperformance problem is defined for an unstable plant. Considerthe family of plants described by the following multiplicativeuncertainty model:

G̃(s) =(s+ 1)(s+ 10)

(s+ 2)(s+ 4)(s− 1)[1+W2(s)∆(s)] (34)

where

W2(s) = 0.81.1337s2 + 6.8857s+ 9

(s+ 1)(s+ 10). (35)

The nominal performance is defined by ‖W1S‖∞ < 1 with:

W1(s) =2

(20s+ 1)2. (36)

The objective is to compute a controller K(s) that optimizes therobust performance by minimizing γ in (32).The standard H∞ solution solves an approximate problem and

leads to γopt = 0.844 with the controller K∞(s) = N∞/D∞, where

N∞ = 7.409e6s6 + 1.266e8s5 + 6.335e8s4 + 1.152e9s3

+ 6.911e8s2 + 5.442e7s+ 9.37e5

D∞ = s7 + 9.07e5s6 + 1.901e7s5 + 1.043e8s4

+ 4.416e7s3 − 4.682e7s2 − 4.962e6s− 1.262e5.This 7th-order controller is unstable and has a pair of complexconjugate poles very close to the imaginary axis.Now, the proposed method is applied to design a PID controller

represented by:

K(s) = [Kp, Ki, Kd][1,1s,

s1+ Tf s

]Twhere the time constant of the derivative part of the PID controllerTf is set to 0.01 s. The frequency response of themodel is computedat N = 500 linearly spaced frequency points between 10−3 and103 rad/s. The uncertainty circle at each frequency is approximatedby a polygon with q = 8 vertices. The plant model contains oneunstable pole and the controller an integrator, so the desired open-loop transfer function is chosen as:

Ld(s) = βs+ αs(s− 1)

. (37)

This is the simplest choice of Ld(s) that contains a stable zero to en-sure the Nyquist stability criterion. The characteristic polynomialof the closed-loop system with Ld(s) is given by: s2 − s + βs +βα. Taking α = 1 for simplicity, the stability criterion is satis-fied for Ld(s) with β > 1. For instance, we choose β = 2 andwill study later the sensitivity of the solution for different valuesof β .In order to obtain the controller giving the minimal value for γ ,

the bisection algorithm explained in Remark (4) is used with thelinear constraints in (11) that leads to‖ |W1S| + |W2T | ‖∞ = 0.7262.

The resulting PID controller is:

K0(s) =2.074s2 + 9.702s+ 6.425

0.01s2 + s. (38)

It is interesting to observe that this PID controller gives betterperformance than the H∞ controller. Moreover, it is stable andeasily implementable on a real system. The performance can befurther improved using a new Ld(s) based on K0(s). With this newLd(s) = K0(s)G(s) the optimal controller is given by:

K(s) =2.643s2 + 23.500s+ 8.589

0.01s2 + s(39)

which leads to γopt = 0.7247.In order to study the sensitivity of the solutions to the choice

of Ld(s), the value of β in (37) is changed from 2 to 97 with astep size of 5. For each value of β the minimum of γ is computed.The mean value of optimal γ ’s is 0.7611 and its standard deviation0.0394. This shows that although the optimal solution depends onthe choice of Ld(s), it is not very sensitive to this choice. Moreover,the results obtained by this approach, whatever the choice ofβ between 2 and 97, are better than the standard H∞ optimalsolution.

5. Conclusions

It should be mentioned that the problem of robust fixed-ordercontroller design is a non-convex NP-hard problem and all solu-tions to this problem, including ours, are based on some approx-imations. It is too difficult (if not impossible) to compare, by atheoretical analysis, the overall approximation or conservatism ofdifferent approaches to fixed-order controller design. In this paperwe tried to show the effectiveness of the proposed approach bymeans of a simulation example. This approach has been applied toan international benchmark problem for robust controller design(Landau, Rey, Karimi, Voda, & Franco, 1995) and a controller withonly 7 parameters has been designed that meets all benchmarkspecifications. These results are not included in this paper becauseof space limitation but are available in Galdos and Karimi (2009).The method uses only the frequency response of the system

and no parametric model is required. The frequency response ofthe model and the uncertainty at each frequency can be obtaineddirectly by discrete Fourier transform from a set of data, so themethod can be considered as completely ‘‘data-driven’’. Of course,the method can be applied as well if a parametric model with apure time delay and an uncertainty set is available.Although only SISO systems are discussed in this paper, the

extension to MIMO systems is also possible thanks to Gershgorinbands. In the same framework, multivariable controllers can bedesigned that decouple the off-diagonal elements of the open-looptransfer matrix and meets the H∞ specifications for the decoupledsystem (Galdos, Karimi, & Longchamp, 2009).

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Alireza Karimi received his DEA and Ph.D. degrees onAutomatic Control from InstitutNational Polytechnique deGrenoble (INPG) in France in 1994 and 1997, respectively.He was Assistant Professor at the Electrical EngineeringDepartment of Sharif University of Technology in Teheranfrom 1998 to 2000. He is currently Senior Scientist atthe Automatic Laboratory of Ecole Polytechnique Fédéralede Lausanne (EPFL), Switzerland. He is Associate Editorof European Journal of Control since 2004. His researchinterests include closed-loop identification, data-drivencontroller tuning approaches and robust control.

Gorka Galdos received his B.Sc. and M.Sc. degrees in Me-chanical Engineering from Ecole Polytechnique Fédéralede Lausanne (EPFL), Switzerland in 2003 and 2005, re-spectively. He is nowworking towards the Ph.D. degree inAutomatic Control at EPFL. His current research is robustcontrol using nonparametric models.