GENETIC SOLUTIONS TO MIXED H

2

/H

∞

PROBLEMS

Limits of Performance

Gustavo S´anchez, Miguel Strefezza

Universidad Sim´on Bol´ıvar, Departamento de Procesos y Sistemas, Caracas, Venezuela

Minaya Villasana

Universidad Sim´on Bolivar, Departamento de C´omputo Cient´ıﬁco, Caracas, Venezuela

Keywords:

Multi-objective control, Genetic algorithms, LMIs, Pole placement, COMPl

e

ib.

Abstract:

One of the most relevant problems for control engineers is the so-called “mixed H

2

/H

∞

”. To solve it, different

convexifying strategies became popular in the later 1990s, mainly based on Linear Matrix Inequalities (LMIs).

On the other hand, genetic algorithms have also been applied for H

2

/H

∞

synthesis. Indeed, several authors

agree that they are able to ﬁnd good solutions to this important control problem. However, in most of the

published papers, only low-order SISO models have been considered. In the present paper a LMI-based

algorithm is compared against a genetic algorithm, with respect to three performance indicators: Set Coverage,

Maximum Distance and Efﬁcient Set Spacing. Five open-loop MIMO models extracted from COMPl

e

ib are

studied, for which the degree varies between 5 and 10. Based on numerical results, the genetic algorithm is

not able to improve LMI solutions for problems with more than 42 variables, restricted to a budget of 20.000

function evaluations.

1 INTRODUCTION

One of the most important problems for control engi-

neers is the so-called “mixed H

2

/H

∞

”. Typically, the

H

∞

channel is used to enhance the robustness of the

closed-loop system, whereas the H

2

channel guaran-

tees good performance (Apkarian et al., 2008).

To solve this problem, different convexifyng

strategies became popular in the later 1990s, despite

the inherent conservatism of this approach. For

instance, in (Scherer et al., 1997) controllers are

designed by solving a set of LMIs in tandem with

nonlinear algebraic equalities. In fact, this design

method (enhanced with many improvements over the

years) remains as state of the art for this problem.

On the other hand, Multi-Objective Evolutionary

Algorithms (MOEA) have also been applied for

H

2

/H

∞

synthesis: in (Takahashi et al., 2001) and

(Takahashi et al., 2004), a genetic approach is pro-

posed to obtain H

2

/H

∞

solutions which are consis-

tent with a Pareto set and less conservative compared

to LMI solutions.

After these examples it may seen obvious that, un-

der special circumstances, genetic algorithms are in

fact able to ﬁnd better solutions than LMI-based al-

gorithms. However, in most of the published works

which have been consulted for this paper, only low-

order and SISO models have been considered, more

appropriate to evaluate low-complexity controllers as

PIDs (Astrom et al., 1998).

In this manner, the question arises as to whether

the genetic algorithm advantage remains true when

the open-loop models are high-order and MIMO

(Multiple Input Multiple Output) as those proposed

in COMPl

e

ib (Leibfritz, 2004).

The rest of this paper is organized as follows. In

section 2, the controller design problem is formulated.

Next, the two design methods to be compared are

described in section 3. In section 4, numerical results

are presented and conclusions are given in section 5.

2 PROBLEM FORMULATION

The closed-loop system is shown in ﬁgure 1. Matri-

ces A ∈ R

n×n

, B ∈ R

n×n

u

and C ∈ R

n

y

×n

denote the

corresponding open-loop state matrices.

The open-loop state-space equations are:

282

Sánchez G., Strefezza M. and Villasana M..

GENETIC SOLUTIONS TO MIXED H2/H∞ PROBLEMS - Limits of Performance.

DOI: 10.5220/0003640702820285

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 282-285

ISBN: 978-989-8425-83-6

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Continuous-time closed-loop design model.

˙x = Ax+ Iw+ Bu

z

1

= y = Cx

z

2

= Ax

(1)

In this formulation, w ∈ L

n

w

×1

2

denotes the exoge-

nous input, z

1

∈ L

n

z

1

×1

2

and z

2

∈ L

n

z

2

×1

2

represents the

outputs to be regulated, while u ∈ L

n

u

×1

2

and y ∈ L

n

y

×1

2

represent the control input and the measured out-

put respectively. It is assumed the open-loop model

is strictly proper, stabilizable from u and detectable

from y.

Consider a full-order linear controller K

c

described by the state equations

K

c

:

˙x

c

= A

K

x

c

+ B

K

y

u = C

K

x

c

+ D

K

y

(2)

Finally, let

G

1

(K

c

) = G

z

1

w

(K

c

)

G

2

(K

c

) = G

z

2

w

(K

c

) (3)

be the closed-loop transfer function from w to z

1

and

z

2

respectively. The mixed H

2

/H

∞

control problem is

stated as:

P

H

2

/H

∞

: min

K

c

kG

1

(K

c

)k

2

kG

2

(K

c

)k

∞

subject to

G

1

(K

c

) and G

2

(K

c

) are stable

(4)

In this formulation the term min should be interpreted

as the search for the best possible approximation of

the corresponding Pareto-optimal set.

3 DESIGN METHODS

In this section two methods to solve the multi-

objective problem P

H

2

/H

∞

are described.

3.1 LMI-based Method

In this sub-section a design algorithm based on the re-

sults presented in (Scherer et al., 1997) is proposed.

In fact, these authors demonstrated that given two

positive scalars γ

2

,γ

∞

the following equations hold

kG

1

(K

c

)k

2

≤ γ

2

(5)

kG

2

(K

c

)k

∞

≤ γ

∞

if there exist matrices X > 0,Y > 0,

b

A,

b

B,

b

C,

b

D such

that a certain set of LMIs are feasible.

To build the approximation of the Pareto-front,

the following iterative procedure is proposed (see

Algorithm 1): First the limit of one restriction is

increased and the other one decreased (i.e. linearly).

Then, the feasability problem is solved and the solu-

tion K

c

is archived, only in case it is non-dominated.

Otherwise it is rejected.

Algorithm 1: Algorithm to solve P

H

2

/H

∞

via LMIs.

Data: G

1

,G

2

,N

H

2

/H

∞

,∆γ

2

,∆γ

∞

,γ

min

,γ

max

Result: P

k = 1;

while k ≤ N

H

2

/H

∞

do

Solve

G

2

(K

k

c

)

2

≤ γ

min

+ k∆γ

2

G

1

(K

k

c

)

∞

≤ γ

max

− k∆γ

∞

P

k

= U pdateArchive(K

k

c

,P

k−1

);

k = k + 1;

end

3.2 Multi-objective Pole Placement with

Evolutionary Algorithms

(MOPPEA)

In the following we describe a design method, named

Multi-Objective Pole Placement with Evolutionary

Algorithms (MOPPEA).

In the general case, an output feedback controller

can be designed by combining a full information con-

troller with a state observer. The resulting output

feedback sub-system is called “observer-based con-

troller” and has the following state-equations:

·

x

c

= (A + BK + LC)x

c

− Ly

u = Kx

c

(6)

where x

c

is the estimated state.

Let pk ∈ C

n

k

and pl ∈ C

n

l

be the eigenvalues of

A + BK and A + LC respectively. To assure closed-

loop system stability, the gain matrix K and L must be

calculated in such way that pk and pl belong to C

−

(open left-half complex plan).

GENETIC SOLUTIONS TO MIXED H2/H8 PROBLEMS - Limits of Performance

283

Thus, the key concept of the proposed design

method is using an evolutionary process in order to

evolve matrices K ∈ R

n

u

×n

and L ∈ R

n×n

y

. Thus, the

mixed H

2

/H

∞

control problem is stated again as:

b

P

H

2

/H

∞

: min

K∈R

n

u

×n

,L∈R

n×n

y

kG

1

(K,L)k

2

kG

2

(K,L)k

∞

subject to

G

1

(K,L) , G

2

(K,L) are stable

(7)

Regarding the initial population, it can be

generated using the algorithm proposed in (S´anchez

et al., 2007). After that, SPEA2 is used to drive

the design process, taking advantage of its ability to

manage an archive of non-dominated solutions.

4 NUMERICAL RESULTS

In this section two algorithms are compared:

A1: SPEA2 - enhanced with special operators and

restricted to 20.000 objective function evalua-

tions, using the parameters shown in table 1.

This quantity was ﬁxed considering the total time

available for computations.

A2: LMI-based design (see Algorithm 1).

Table 1: Setting parameters used for SPEA2.

Parameter Value

Initial Population Randomly generated

Representation K + L

Cross-Over Recombination Arithmetical

Cross-Over Rate 0.9

Mutation Operator Gaussian

Mutation Rate 0.1

Population Size 200

Stop Condition 100 generations

Population Size 100

Offspring Size 100

Table 2 (at the top of the next page) presents

the information related with the selected COMPl

e

ib

models, each one characterized by a particular

nomenclature. Five models were selected: AC1, AC6,

WEC1, NN10 and AC9, for which the number of

decision varies between 30 and 90. Table 2 also

presents, for each model, the parameters N

H

2

/H

∞

,∆γ

2

,

∆γ

∞

, γ

min

and γ

max

used by A2.

Thirty executions were simulated for each algo-

rithm and for each problem. Let P F

1

and P F

2

be

the Pareto approximations found by two different al-

gorithms. To compare their performance, the follow-

ing indicators were computed:

• Set Coverage(C)

• Maximum Distance(MD)

• Efﬁcient Set Spacing(ESS)

The values obtained for these indicators are shown

in tables 3 and 4. For each indicator the mean value

and the standard deviation within parentheses are pre-

sented. These results conﬁrm that the LMI-based

algorithm is able to produce dominating solutions

with respect to the genetic algorithm, for problems

with more than 42 decision variables. Note that A1

achieves better results than A2 with respect to MD.

However, A2 achieves better results with respect to

ESS, which can be explained given the deterministic

nature of A1.

Table 3: Set coverage results.

C(A

i

,A

j

) A1 A2

A1 −

AC1:0.8619(0.1326)

AC6:0(0)

NN10:0(0)

WEC1:0(0)

AC9:0(0)

A2

AC1:0(0)

AC6:0(0)

NN10:1(0)

WEC1:1(0)

AC9:1(0)

−

Table 4: MD and ESS results.

MD ESS

A1

AC1:0.7589(0.0535)

AC6:120.4263(67.0600)

NN10:3.5940(0.7046)

WEC1:45.3320(7.7541)

AC9:564.2026(190.4456)

AC1:0.0190(0.0069)

AC6:3.5698(4.0164)

NN10:0.1026(0.0407)

WEC1:1.0784(0.4881)

AC9:21.5037(17.8440)

A2

AC1:0.1618(0)

AC6:2.8065(0)

NN10:1.0200(0)

WEC1:36.0976(0)

AC9:59.2017(0)

AC1:0.0122(0)

AC6:0.0152(0)

NN10:0.0188(0)

WEC1:0.1540(0)

AC9:3.3800(0)

5 CONCLUSIONS

In this paper, we analyze the performance of a genetic

algorithm (SPEA2) to solveﬁve mixed H

2

/H

∞

design

problems, taking as reference a LMI-based iterative

algorithm and considering a ﬁxed budget of 20.000

evaluations.

Based on the obtained results, the following con-

clusions can be stated:

• Unlike other representations, the proposed (K,L)

chromosome is able to efﬁciently explore the con-

troller space, even for models with order greater

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

284

Table 2: Information related with the selected COMPl

e

ib models.

COMPl

e

ib n n

u

n

y

n

MOPPEA

N

H

2

/H

∞

∆γ

2

∆γ

∞

γ

min

γ

max

r

max

i

max

AC1 5 3 3 30 100 0.01 0.01 1.5 5.5 10 10

AC6 7 2 4 42 100 0.1 1 1 110 20 20

NN10 8 3 3 48 100 0.01 0.02 3 15 10 10

WEC1 10 3 4 70 100 0.05 1 2 200 100 100

AC9 10 4 5 90 100 0.1 0.1 50 100 100 100

than 4 and with multiple inputs and outputs. The

proposed variation operators allows to stay within

the feasible region.

• The statistical tests show the genetic algorithm is

not able to improve LMI solutions for problems

with more than 42 variables, considering a ﬁxed

budget of 20.000 function evaluations.

As future work, more simulations need to be

carried out, to ﬁnd how many function evaluations

are to be allowed in order the genetic algorithm is

competitive again. It is also possible to test ”hy-

brid” design methods, based on both deterministic

and stochastic strategies to ﬁnd better Pareto approxi-

mations.

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H

2

/H

∞

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Leibfritz, F. (2004). COMPlib - Constrained Matrix-

optimization Problem Library. Technical report, Uni-

versity of Trier. Department of Mathematics, Ger-

many.

S´anchez, G., Villasana, M., and Strefezza, M. (2007).

Multi-Objective Pole Placement with Evolutionary

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