H2/H∞ Robust PID Synthesis for Uncertain Systems

E. N. Goncalves, R. M. Palhares, R. H. C. Takahashi

Abstract— This paper presents a procedure for robust PIDtuning based on a non-convex optimization problem with H2

and/or H∞ performance criteria and regional pole placement.The uncertain system can be represented by polytopic or affineparameter-dependent models. The efficiency of the proposedtuning procedure is illustrated by two examples.

Index Terms— PID tuning, polytopic models, affineparameter-dependent models, non-convex optimization.

I. INTRODUCTION

The proportional-integral-derivative (PID) controller re-mains the most applied controller configuration in the in-dustry. More than 90% of all control loops are PID with awide range of applications: process control, motor drives,magnetic and optic memories, automotive, flight control,instrumentation, etc. [1]. This fact is probably due its sim-plicity (only three parameters to adjust), effectiveness for themajority of the industrial plants and the existence of simpletuning methods. In 1942, John G. Ziegler and NathanielB. Nichols developed a step response method in which thePID parameters are computed from simple features of thestep response but it is known to give very poor results inmany cases [1]. After that, several works have been proposedto meet specific requirements according to the applicationconsidering quite different objectives and tuning algorithmcomplexity (see the references in [1]). An overview of PIDcontrol and different tuning methods are presented in [2]. TheZiegler-Nichols step response method is revisited and newtuning rules are developed in [3] and [4]. The design of PIDcontrollers based on constrained optimization is developedin [5] where the objective is to maximize the integral gainsubject to constraints over the Nyquist curve. In [6], a singlePID tuning rule for a first-order or second-order time delaymodel are presented and simple analytic rules for modelreduction are developed to put an original system into therequired model.

Despite the several PID tuning techniques available fordifferent PID configurations and applications, it is still inter-esting to research new approaches that can lead to betterperformance and that can be applied for a broader class

This work has been supported in part by CNPq and FAPEMIG, Brazil.E. N. Goncalves is with Department of Electrical Engineering,

Federal Center of Technological Education of Minas Gerais,Av. Amazonas 7675, 31510-470, Belo Horizonte, MG, Brazileduardong@des.cefetmg.br.

R. M. Palhares (corresponding author) is with Department ofElectronics Engineering, Federal University of Minas Gerais, Av.Antonio Carlos 6627, 31270-010, Belo Horizonte, MG, Brazilpalhares@cpdee.ufmg.br.

R. H. C. Takahashi is with Department of Mathematics, Federal Univer-sity of Minas Gerais, Av. Antonio Carlos 6627 - 31270-010, Belo Horizonte,MG, Brazil taka@mat.ufmg.br.

of problems, in special, the multiobjective robust synthesisto tackle uncertain systems represented by polytopic oraffine parameter-dependent models. Linear matrix inequal-ities (LMIs) may be one of the alternatives to develop PIDtuning approaches to be applied to uncertain systems. Onthe other hand, the PID tuning may be put as a staticoutput-feedback control leading to bilinear matrix inequal-ities (BMIs) that are non-convex optimization problem. In[7], the PID tuning is characterized as a robust multiobjectivestatic output-feedback control problem with an augmentedstate equations and LMI-based formulations are applied toassess the resulting guaranteed costs. In [8], the Kharitonovtheorem for interval plants is applied for the purpose ofcharacterizing all PID controllers that stabilize an uncertainplant. In [9], a robust PID controller design method formultiple-models is developed via the LQR-LMI static output-feedback control approach.

This paper presents a robust PID tuning procedure foruncertain systems based on two stages. In the first stage,an non-convex optimization problem is solved to computethe PID parameters considering a finite set of points of theuncertainty domain. In the second stage, a branch-and-boundalgorithm combining LMI sufficient conditions and polytopepartition is applied to evaluate the objective function andconstraints for all uncertainty domain. The second stageis required in order to identify the necessity of includingadditional points in the finite set considered in the first stage,for a next iteration. Similar procedure has been proposedin the context of robust state-feedback controller synthesis[10], robust static and dynamic output-feedback controllersynthesis [11], and filter synthesis [12]. Despite the difficultto tackle non-convex optimization problems with only threedecision variables (the PID parameters), it will be illustratedby examples that the proposed robust PID tuning procedurecan provide good closed-loop system performance.

II. PROBLEM STATEMENT

This paper considers the two-degrees-of-freedom ISA PIDconfiguration presented in Fig. 1 where R(s) is the set point,D(s) is the load disturbance, U(s) is the control signal, C(s)is the system output, and η(s) is the measurement noise. TheISA PID control law is given by

U(s) = −kp

(Tis + 1

Tis+

Tds

ρTds + 1

)[C(s) + η(s)]

+kp

Tis + 1

TisR(s)

(1)

Proceedings of the 45th IEEE Conference on Decision & ControlManchester Grand Hyatt HotelSan Diego, CA, USA, December 13-15, 2006

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G(s)K

T sT s

R(s) U(s)E(s)

η(s)

C(s)

i

p

d

1

ρT s+1d

D(s)

+

+

+

+

++

+

+

- -

G (s)d

Fig. 1. Block diagram of the ISA PID configuration.

where kp is the proportional gain, Ti is the integral actiontime or reset time, Td is the derivative action time or ratetime, and N 1/ρ is the noise filtering constant that variesbetween 3 and 10 (see [3]) with N = 10 the typical value.

In the PID controller tuning, the resulting system mustpresent the following features: good set point tracking,satisfactory load disturbance rejection, minimal influence ofthe measurement noise, limited range and rate of variationof the control signal, and robustness to model uncertainties.There are different ways to enforce these requirements inthe PID controller synthesis strategy. For example, the loaddisturbance rejection can be achieved by the minimizationof suitable closed-loop transfer-function norms, and require-ments over time responses can be attained by regional poleplacement constraints. To achieve a robust PID controller,the tuning approach must have the capability to deal withuncertain systems represented, for instance, by polytopic oraffine parameter-dependent models.

Fig. 2 presents a general block diagram of the output-feedback control system. The linear time-invariant systemP (s) can be represented in the state-space as

x(t) = Ax(t) + Bww(t) + Buu(t)

z1(t) = Cz1x(t) + Dzw1w(t) + Dzu1u(t)

z2(t) = Cz2x(t) + Dzw2w(t) + Dzu2u(t)

y(t) = Cyx(t) + Dyww(t) + Dyuu(t)

(2)

where x ∈ Rn is the state space vector, w(t) =

[r(t), d(t), η(t)]T is the exogenous input vector, u(t) ∈ R

is the control variable, z1(t) ∈ Rq1 , and z2(t) ∈ R

q2 arethe controlled output vectors associated with the H∞ andH2 performance indices, respectively, and y(t) = [c(t) +η(t), r(t)]T is the output vector.

This work is concerned with PID tuning for uncertainsystems where the matrices of the state-space model are notprecisely known. Define the system matrix as

P

⎡⎢⎢⎣A Bw Bu

Cz1 Dzw1 Dzu1

Cz2 Dzw2 Dzu2

Cy Dyw Dyu

⎤⎥⎥⎦

y

z

z

u

wP(s)

K(s)

1

2

Fig. 2. General block diagram of the closed-loop control system.

In the case of polytopic models, the system matrix be-longs to a bounded polyhedric convex domain, or polytope,described by the convex combination of its N vertices:

P (θ) =

N∑i=1

θiPi (3)

where

Pi

⎡⎢⎢⎣Ai Bw,i Bu,i

Cz1,i Dzw1,i Dzu1,i

Cz2,i Dzw2,i Dzu2,i

Cy,i Dyw,i Dyu,i

⎤⎥⎥⎦ (4)

are the set of known vertices and θ = [θ1 . . . θN ]T ∈ ΩM isthe polytopic coordinate vector, with ΩM defined as

ΩM

θ : θi > 0, i = 1, . . . , N,

N∑i=1

θi = 1

(5)

In the case of uncertain models depending affinely on anuncertain parameter vector p = [p1 . . . pd]

T , where pi haslower and upper bounds, p

iand pi, respectively, the system

matrix is computed as

P (p) = P0 +

d∑i=1

piPi (6)

with Pi, i = 0, . . . , d, as in (4).In this case, p ∈ Ωp, where Ωp is a polytope with the

shape of an hyper-rectangle, with the vertices generated bythe combination of the uncertain parameter bounds, or any

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other shape if there are additional linear constraints over theuncertain parameters.

The ISA PID controller, K(s) in Fig. 2, can be describedby the following realization considering the state variablesxc1(t) e xc2(t) of the modified block diagram presented inFig. 3:

K(s) =

[Ac Bc

Cc Dc

]

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣0 0 −

1

Ti

1

Ti

0 −1

ρTd

1

ρ2Td

0

kp kp −kp

(1 +

1

ρ

)kp

⎤⎥⎥⎥⎥⎥⎥⎥⎦(7)

K

1/ρ

T sR(s)

U(s)

E(s)

η(s)

C(s)

i

p

1

ρT s+1d+

+

+

++

+

-

X (s)

X (s)

c2

c1

-

+

1/ρ

K(s)

+

Fig. 3. Modified ISA PID block diagram.

Let α denotes θ or p and Ω denotes ΩM or Ωp. The closed-loop system is given by

Tziw(s, α) =

⎡⎣ A(α) Bw(α)

Ci(α) Di(α)

⎤⎦

=

⎡⎣ A + BuDcCy BuCc

BcCy Ac

Czi + DzuiDcCy DzuiCc

∣∣∣∣∣∣∣∣∣∣∣∣

Bw + BuDcDyw

BcDyw

Dzwi + DzuiDcDyw

⎤⎦ , i ∈ 1, 2

(8)Consider D ∈ C− the desired closed-loop pole placement

region (half-plane, disk, sector, or the intersection of them)with C− denoting the left half-plane. Let γw.c. and δw.c. bethe worst case of the H∞ and H2 norms in the uncertaintydomain:

γw.c. maxα∈Ω

‖Tz1w(s, α)‖∞ (9)

δw.c. maxα∈Ω

‖Tz2w(s, α)‖2 (10)

The robust PID tuning problem can be stated as: find thevalues of kp, Ti, and Td that minimizes the maximum norm

values γw.c. and δw.c. subject to kp > 0, Ti ≥ 0, Td ≥ 0, andλi(A(α)) ∈ D,∀i,∀α ∈ Ω (λi denotes the i−th eigenvalue).

The regional pole-placement constraint is useful becausethe optimal H∞ controllers may lead to slow set pointresponses [7].

III. PID TUNING PROCEDURE

Let Ω ⊂ Ω be a finite set of polytope points, initialized asthe set of polytope vertices Ω = ν(Ω), where ν(·) denotesthe set of vertices of its argument. Consider the “worst-case”points in the finite set Ω, given a PID controller K(s):

γw.c. = maxα∈eΩ

‖Tz1w(s, α)‖∞

δw.c. = maxα∈eΩ

‖Tz2w(s, α)‖2(11)

Let γg.c. and δg.c. be the ε-guaranteed costs such as

γw.c. ≤ γg.c. ≤ (1 + ε)γw.c. (12)

δw.c. ≤ δg.c. ≤ (1 + ε)δw.c. (13)

and α(∞) ∈ Ω and α(2) ∈ Ω be the “worst case” coordinates:

α(∞) arg maxα∈Ω

‖Tz1w(s, α)‖∞ (14)

α(2) arg maxα∈Ω

‖Tz2w(s, α)‖2 (15)

Define the set of PID controllers, as in (7), that satisfiesthe regional pole-placement constraints:

Γ

K(s) : kp > 0, Ti ≥ 0, Td ≥ 0, ρ = 0.1

λi(A(α)) ⊂ D, ∀i, ∀ α ∈ Ω

(16)

The following auxiliar problem is considered in the pro-posed PID tuning procedure:Auxiliar Problem: Given the scalars λ1 > 0, λ2 > 0, λ1 +λ2 = 1, ε1 > 0, and ε2 > 0, find kp, Ti, and Td to constructK∗(s), such as:

K∗(s) = arg minK(s)

maxα

(λ1‖Tz1w(s, α)‖∞

+λ2‖Tz2w(s, α)‖2)

subject to:

⎧⎪⎪⎨⎪⎪⎩α ∈ Ω

K(s) ∈ Γ‖Tz1w(s, α)‖∞ ≤ ε1‖Tz2w(s, α)‖2 ≤ ε2

The proposed iterative robust PID tuning procedure, to bepresented in the sequel, has two stages. At each iteration,in the synthesis stage, the PID parameters are computed bymeans of an optimization algorithm considering the finite setof polytope points Ω. In the analysis stage, considering thepolytope Ω, if the maximum value of the objective functionoccurs in a point outside the finite set Ω or if any constraintis not attained in a point belonging to the polytope Ω,then these points are included in the finite set and anotheroptimization stage is processed. The procedure concludeswhen all constraints are attained for the whole polytope andthere is no possibility to minimize the objective function by

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means of the introduction of new points in the finite setsconsidering a relative accuracy index εδ .PID Tuning Procedure:

Step 1. Initialize i ← 0, Ω0 ← ν(Ω).Step 2. i ← i + 1, Ωi ← Ωi−1.Step 3. Solve the auxiliar problem to compute K∗(s) and

γw.c. and/or δw.c.

Step 4. If λ1 > 0 or ε1 < ∞ then compute γg.c. andα(∞) for K∗(s).If α(∞) ∈ Ωi, then if λ1 > 0 and (γg.c. −γw.c.)/γw.c. > εδ or γg.c. > ε1, thenΩi ← Ωi ∪ α(∞).

Step 5. If λ2 > 0 or ε2 < ∞ then compute δg.c. and α(2)

for K∗(s).If α(2) ∈ Ωi, then if λ2 > 0 and (δg.c. −

δw.c.)/δw.c. > εδ or δg.c. > ε2, thenΩi ← Ωi ∪ α(2).

Step 6. Verify if ∃ i | λi(A(α(u))) ∈ D for any α(u) ∈ Ω.If true, then Ωi ← Ωi ∪ α(u).

Step 7. If Ωi = Ωi−1, then goto step 2, else finalize.

A. Optimization Stage

The auxiliar scalar optimization problem can be solvedby means of the cone-ellipsoidal algorithm [13]. Con-sider the ellipsoid in the iteration k described as Ek =x ∈ R

d | (x − xk)T Q−1k (x − xk) ≤ 1

, where xk is the

ellipsoid center and Qk = QTk 0 is the matrix that

determines the direction and the dimension of the ellipsoidaxes. Given the initial values x0 and Q0, the ellipsoidalalgorithm is described by the following recursive equations:

xk+1 = xk −1

d + 1Qkm

Qk+1 =d2

d2 − 1

(Qk −

2

d + 1QkmmT Qk

) (17)

withm = mk/

√mT

k Qkmk.

where x ∈ Rd is the vector of optimization parameters (in

this case the PID parameters) and mk is the gradient (or sub-gradient) of the most violated constraint of g(x) : R

d → Rs,

when xk is not a feasible solution, or the gradient (or sub-gradient) of the objective function, f(x) : R

d → R, when xk

is a feasible solution. The optimization algorithm ends when(fmax − fmin)/fmin ≤ ε, where fmax and fmin are theinferior and superior values of the objective function at thelast Nε iterations and ε is the prescribed relative accuracy.

B. Analysis Stage

The computation of the ε-guaranteed costs (γg.c or δg.c.),maximum value of the closed-loop transfer function norms(γw.c or δw.c.) for all uncertain domain, and the correspon-dent coordinates (α(∞) or α(2)) is performed by means of ananalysis approach based on the branch-and-bound algorithm.The basic strategy of this algorithm is to partition theuncertainty domain such as lower and upper bound functions

converge to the maximum value of the norm. This algorithmends when the difference between the bound functions islower than the prescribed relative accuracy ε. The algorithmis implemented considering as lower bound function theH∞ (or H2) norm computed in the vertices and as upperbound function the H∞ (or H2) guaranteed cost computed bymeans of LMI-based formulations, both functions calculatedfor the original polytope and its subdivisions [14], [15].An partition technique based on simplicial meshes [16] isapplied to tackle uncertainty domains not restricted to thehyper-rectangle case. This partition technique allows thisalgorithm be applied to both affine parameter-dependentas well polytopic models with improved efficiency. Thesame partition strategy is also applied to the regional poleplacement analysis where if the system is not robustly D-stable, the analysis procedure find and return the coordinateα(u) of one instance of a not D-stable model Tziw(s, α(u))such as ∃ i | λi(A(α(u)) ∈ D (see [17]).

IV. ILLUSTRATIVE EXAMPLES

The results presented in the following examples werecomputed with the software MATLAB in a computer withan 2.8Ghz processor and 1Gb system RAM.

Example 1 - Consider the uncertain system presented in [8]:

G(s) =5.2(s + 2)

s(s3 + b2s2 + b1s + b0)

where the coefficients b0, b1, and b2 of the denominator liewithin the following bounds: 9.5 ≤ b0 ≤ 11.5, 12 ≤ b1 ≤15, and 3.5 ≤ b2 ≤ 4.8. In [8], a robust classical PIDis designed to guarantee the gain and phase margins. Thefollowing realization is considered here with the definitionof the of the controllable variables, and the addition of aninput disturbance, Gd(s) = G(s), and measurement noise(see Fig. 1):

d

dt

264

x1

x2

x3

x4

375 =

264

0 0 0 01 0 0 −b0

0 1 0 −b1

0 0 1 −b2

375

264

x1

x2

x3

x4

375

+

264

0 10.4 00 5.2 00 0 00 0 0

375

24 r

dη

35 +

264

10.45.200

375 u

z1 =

»0 0 0 10 0 0 0

– 264

x1

x2

x3

x4

375 +

»01

–u

y =

»0 0 0 10 0 0 0

– 264

x1

x2

x3

x4

375 +

»0 0 11 0 0

– 24 r

dη

35

The proposed PID tuning procedure is applied to computethe controller that minimizes the H∞ guaranteed cost andplace the poles at the intersection of the half-plane Real(λ) <−0.5 and the sector centered at origin with inner angle

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16π/18, ∀p ∈ Ωp, where p = [b0 b1 b2]T and Ωp is a

hyper-rectangle. The proposed procedure computes the PIDparameters as kp = 1.5306, Ti = 2.8413, and Td = 0.3877.The PID parameters kp = 3.1950, Ti = 1.3975, andTd = 0.2236, calculated with the ZieglerNichols frequencyresponse method are considered as initial condition to theoptimization algorithm. The H∞ guaranteed cost is improvedfrom 38.485 to 16.906 after 49s of computational time. Fig. 4presents the time responses of the PID controllers presentedin [8] and computed here for an unity step in r(t), a negativeunit step in d starting from t = 15s and a noise signalvarying randomly in the range |η| ≤ 0.01. The proposedPID controller presents better disturbance rejection and it isless influenced by the uncertain parameter variations.

0 5 10 15 20 25 300

0.25

0.5

0.75

1

1.25

1.5

Time (sec)

Out

put

Proposed (solid)

Huang andWang, 2000 (dashed)

Fig. 4. Time responses of y(t) at the eight vertices of the uncertaintydomain (Example 1).

Example 2 - Consider a continuous stirred tank reactorwith a multiple-model description presented in [9] and alsoconsidered in [18]. Considering an stable operation range,three models are obtained for different operating points:

G1(s) =0.04612

s2 + 9.251s + 22.19

G2(s) =0.04107

s2 + 2.674s + 10.97

G3(s) =0.03707

s2 + 0.01248s + 5.862

These three models are the vertices of the polytope consid-ering a polytopic model of the uncertain system. In [9], theclassical PID controller is designed to minimize a LQR costand to place the closed-loop poles in the intersection of thehalf plane Real(λ) < −1 and the sector region centered atthe origin with inner angle 3π/4. In [9], the PID parametersare computed by means of the LMI approach that results inkp = 516.6, Td = 0.2784, and Ti = 0.6749. In [18], thePID parameters are computed as kp = 698.1, Td = 0.5259,and Ti = 0.6197. In the proposed approach, considering a

input disturbance with Gd(s, θ) = 500G(s, θ) and a noisemeasurement, the three vertices are realized as

A1 =

»0 −22.191 −9.251

–, Bw,1 =

»0 23.06 00 0 0

–,

A2 =

»0 −10.971 −2.674

–, Bw,2 =

»0 20.5350 00 0 0

–,

A3 =

»0 −5.8621 −0.01248

–, Bw,3 =

»0 18.5350 00 0 0

–,

Bu,1 =

»0.04612

0

–, Bu,2 =

»0.04107

0

–,

Bu,3 =

»0.03707

0

–,

Cz1,1 = Cz1,2 = Cz1,3 =

»0 10 0

–,

Dzu1,1 = Dzu1,2 = Dzu1,3 =

»01

–,

Cy,1 = Cy,2 = Cy,3 =

»0 10 0

–,

Dyw,1 = Dyw,2 = Dyw,3 =

»0 0 11 0 0

–.

The proposed PID tuning procedure is applied to computethe controller that minimizes the H∞ guaranteed cost andplace the poles at the intersection of the half-plane Real(λ) <−2.5 and the sector centered at origin with inner angle π/2,∀θ ∈ ΩM . The Ziegler-Nichols step response method com-putes the PID parameters as kp = 2.5209 × 103, Ti = 0.25,and Td = 0.04. Starting from these parameters, the proposedapproach computes the PID parameters as kp = 2.5196×103,Ti = 0.4435, and Td = 0.1490. The optimization improvesthe H∞ guaranteed cost from 1.0326× 106 to 2.7862× 104

after 3.14s of computational time. The time responses of thePID controllers achieved with the LMI approach in [9], thenumerical optimization approach in [18], and the proposedprocedure are presented in Fig. 5 for an unity step in r(t),a negative unit step in d starting from t = 5s and a noisesignal varying randomly in the range |η| ≤ 0.01.

In [9], it is also tackle an unstable operation range,considering three models for different operating points:

G1(s) =0.036

s2 − 0.405s + 4.996

G2(s) =0.026

s2 − 2.647s − 0.879

G3(s) =0.016

s2 − 2.251s − 2.252

The PID parameters kp = 804.6, Ti = 1.3249, andTd = 0.3930 are computed in [9]. The proposed approachcomputes the PID parameters as kp = 2.5189 × 103, Ti =0.5821, and Td = 0.2179. The optimization starts from anot robustly stable closed-loop system and concludes with

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the H∞ guaranteed cost equal to 2.7862×104 after 4.92s ofcomputational time. The time response of the PID controllersachieved in [9] and the one proposed here are presented inFig. 6 for the same input considered previously.

0 1 2 3 4 5 6 7 8 9 100

0.25

0.5

0.75

1

1.25

Time (sec)

Out

put

Proposed (solid)

Ge et. al., 2002 (dashed)

Toscano, 2005 (dash−dot)

Fig. 5. Time responses of y(t) at the three polytope vertices for the stableopen-loop systems (Example 2).

0 1 2 3 4 5 6 7 8 9 100

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

Time (sec)

Out

put

Proposed (solid)

Ge et. al., 2002 (dashed)

Fig. 6. Time responses of y(t) at the three polytope vertices for the unstableopen-loop systems (Example 2).

In this example, in both ranges, the proposed PID con-trollers present better tracking responses and disturbancerejections and they are less influenced by the uncertainparameter variations than the PID controllers stated in [9],[18].

V. CONCLUSION

In this paper has been presented an efficient two-stage (synthesis/analysis) iterative procedure for robusttwo-degrees-of-freedom PID tuning for uncertain lineartime-invariant systems with polytopic or affine parameter-dependent models. The PID parameters are computed bymeans of a non-convex optimization approach to minimize

the H∞ and/or H2 guaranteed cost of closed-loop transfer-functions and to robustly place the closed-loop poles at LMIregions. Examples are presented to illustrate the better per-formance of the proposed procedure in relation to previousrobust tuning methods in relation to the set point tracking,load disturbance rejection, measurement noise attenuation,and robustness to model uncertainties.

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[12] ——, “H2/H∞ filter design for systems with polytope-boundeduncertainty,” IEEE Transactions on Signal Processing, vol. 54, no. 9,pp. 3620–3626, 2006.

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[14] E. N. Goncalves, R. M. Palhares, R. H. C. Takahashi, and R. C.Mesquita, “H2 and H∞ ε-guaranteed cost computation of uncertainlinear systems,” IEE Proceedings - Control Theory & Applications, toappear.

[15] E. N. Goncalves, S. B. Bastos, R. M. Palhares, R. H. C. Takahashi,R. C. Mesquita, C. D. Campos, and P. Y. Ekel, “H∞ guaranteed costcomputation for uncertain time-delay systems,” in Proceedings of theJoint 44th IEEE Conference on Decision and Control and EuropeanControl Conference. Seville, Spain: IEEE, December 2005.

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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThIP11.5

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