AN LMI-BASED GENETIC ALGORITHM FOR GUARANTEED

COST CONTROL

G.A.Papakostas, O.I. Kosmidou and I.E. Antonakis

Democritus University of Thrace, Department of Electrical and Computer Engineering, 67100 Xanthi, Greece

Keywords: Genetic Algorithms, Linear Matrix Inequalities, Robust Control, Guaranteed Cost Control

Abstract: In this paper a new approach for the Guaranteed Cost Control Problem (GCCP) is presented, using two

efficient tools, Linear Matrix Inequalities (LMIs) and Genetic Algorithms (GAs). A linear system with

parametric uncertainty is considered for which a control law is to be found, minimizing a performance

index. In a previous paper, an efficient method has been presented by using an LMI optimisation technique.

A combined use of LMIs and GAs is proposed in the present approach that allows further improvement of

the design procedure.

1 INTRODUCTION

The guaranteed cost control problem (GCCP) has

drawn considerable attention in the last few years

(Kosmidou, 1991), since the inclusion of

uncertainties in the system model, with the

appearance of robust control theory, is a standard

practice. Most of the approaches to this problem

make use of linear optimization techniques to find

best solutions that satisfy some constraints of a

specified problem. For convex problems, LMI

techniques are now very efficient (Fischman, 1996).

However, searching methods have the major

drawback due to the fact that they may converge to a

local minimum or maximum.

Genetic Algorithms (GAs) come to negotiate this

drawback, providing an alternative stochastic

searching process, in which the natural evolutionary

theory is adapted. Successive application of GAs in

control theory has led to very promising results

(Kundu, 1996).

In this paper, a new genetic algorithm is

presented, which makes use of the LMI tool to

compute the fitness value, of its candidate solution

of the current population.

The paper is organized as follows: In Section 2

the guaranteed cost control problem is formulated. A

general presentation of Genetic Algorithms and the

basic operators they used is given in Section 3. The

main idea of the proposed technique is described in

details in Section 4, while an experimental

verification of this method is performed through a

numerical example, in Section 5. Finally, Section 6

presents conclusions and perspectives.

2 PROBLEM FORMULATION

Consider the linear uncertain system in state-space

representation

() ()( )() ()()

tutBBtxtAAtx ∆++∆+=

()

(1)

where x(t)∈R

is the state vector, u(t)∈R

is the

control vector, A

and B

are the state and control

matrices, respectively, having appropriate

dimensions.

The system uncertainties are described by

() () ()

≤=∆

∑

tbBtbtB

taAtatA

(2)

where the scalars a

(t), b

(t) are uncertain parameters,

possibly time-varying, belonging to specified ranges,

and A

, B

are given constant matrices determining

the uncertainty structure. Without loss of generality,

one can always assume that A

, B

have unity rank

327

A. Papakostas G., I. Kosmidou O. and E. Antonakis I. (2004).

AN LMI-BASED GENETIC ALGORITHM FOR GUARANTEED COST CONTROL.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 327-333

DOI: 10.5220/0001145303270333

 SciTePress

and thus they may be decomposed in form of

products of vectors of appropriate dimensions, as

follows:

iii

gfBedA == ;

(3)

This is called the rank-1 decomposition. By

using these vectors, define the matrices

[

]

[

]

[] [ ]

() (

ttdiagTssdiagS

ggGeeE

ffFddD

...:...:

)

(4)

where s

, i=1,…,p and t

, i=1,…,q are positive

scalars. Since decomposition (3) is not unique, these

scalars may be chosen to determine a suitable rank-1

decomposition in order to satisfy different design

requirements. In other words the elements S, T will

be treated as free parameters, in the design

procedure.

Consider also the quadratic performance index of

the form:

()( ) () () () ()

∫

∞

000

][, dttuRtutxQtxtuxJ

(5)

with Q

>0, R

>0 .

As shown in (Kosmidou, 1996, Fischman, 1996)

the guaranteed cost control law of the form

() ()

tPxBRtu

T1* −

−=

(6)

ensures an upper bound of the quadratic

performance index (5) for all parameter variations

consistent with (2), called guaranteed cost,

()

(

)

() ()

PxxJtuJ

=≤

(7)

The nxn matrix P is the positive definite solution

of the modified Riccati equation associated with the

GCC problem,

)

([

111

=++−−

−−+

−

−−−

−

QESEPDSDFTF

BGRTGBR

BBRPPAPA

TTT

has been proposed in

(Fis

thod, by involving genetic algorithms,

for the searching of the free parameters (S, T, δ) of

the system.

, depending on the specific

app

, there are some specific

pro

he sequel, terminology in the field of genetic

me ods for optimization and searching purposes is

giv

is a solution of a problem satisfying

the

cha

ystems. The fitter

(8)

The scalar δ and the scaling matrices S, T are

chosen by the designer.

Since the GCC problem is often related with

conservatism, i.e. the resulting upper bounds are too

large with respect to the minimal J obtained from the

LQR optimal design for the system without

uncertainty, it is desirable to make J

as small as

possible. This leads to an auxiliary optimization

problem, which is often analytically not tractable.

An efficient solution

chman, 1996) by solving an LMI objective

minimization problem.

That approach is being improved using the

proposed me

3 GENETIC ALGORITHMS

Genetic Algorithms (GAs) have played a major role

in many applications of the Engineering Science. As

mentioned above, GAs constitute a powerful tool to

optimization tasks. In other words, a simple GA is a

stochastic method that performs searching in wide

search spaces and depends on some probability

values. For these reasons as well as its parallel

nature, it has the ability to converge to the global

minimum or maximum

lication, and to skip possible local minima or

maxima, respectively.

The main idea in which GAs are based, was first

inspired by J. Holland (Holland, 2001). He tried to

find a method to imitate the evolutionary process

that characterizes the evolution of living organisms.

This theory is based on the mechanism qualified by

the survival of the fittest individuals over a

population. In fact

cedures taking place until the predominance of

the fittest individual.

In t

en:

Individual

constraints and demands of the system in which

it belongs.

Population is a set of candidate solutions of the

problem, which contains the final solution.

Fitness is a real number value, which

racterizes any solution and indicates how proper

is this solution for the problem under consideration.

Selection is an operator applied to the current

population, in a manner similar to the one of natural

selection found in biological s

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

328

indi

d and new

can

sin

ext population, and also an

acc

. According to this

pro

ue until a predefined

num

imple Genetic Algorithm, which uses some

of the operations discussed above, is presented in

Fig.1.

viduals are promoted to the next population and

poorer individuals are discarded.

Crossover is the second operator that follows the

previous one. This operator allows solutions to

exchange information, in such a way that the living

organisms use in order to reproduce themselves.

More specifically, two solutions are selected to

exchange their sub-strings from a single point and

after (single point crossover), according to a

predefined probability P

. The resulting offsprings

carry some information from their parents. In this

way new individuals are produce

Initial

population

Decode

the

population

Find

Fitness

Selection

Crossover

MutationReplace

New

population

didate solutions are tested in order to find the one

that satisfies the appropriate objective.

Mutation is the third operator that can be applied

to an individual. According to this operation each

Figure 1: Block diagram of a simple GA

In the next section, an LMI-based genetic

algorithm is proposed and applied to the guaranteed

cost control problem described in Section 2.

gle bit of an individual binary string, can be

flipped with respect to a predefined probability P

There is a different procedure that can be

considered for a single iteration of a genetic

algorithm, called Elitism. During this operation the

probability of discarding the fittest individual is

minimized, since at each generated population, the

fittest individual is checked whether it has a lower

fitness than the elite member of the previous

generation. If so, a randomly selected individual is

replaced by the old elite member. Thus, it is

guaranteed that the fittest individual will be

promoted to the n

4 LMI-GENETIC METHOD

As previously mentioned, a first attempt of using

LMI optimizations in Guaranteed Cost Control

(Fischman, 1996) appeared very promising. Besides,

genetic algorithms seem to be very efficient in

solving various optimization problems in which the

searching space is complex.

In the proposed method, a combination of two

optimization tools is used. More precisely, the

genetic algorithm is used to find a suitable variable

set (S, T, δ), while the LMI optimization, used in

(Fischman, 1996) is applied to find an optimum

matrix P that satisfies some prespecified constraints.

By using the obtained matrix P, a fitness value is

assigned to the corresponding candidates and

process goes on.

eleration of the overall speed of the algorithm is

derived in this way.

In the present algorithm, an elitism based

reinsertion method is used

cess, a predefined number of individuals (the

least suitable) are replaced.

Since these operators have been applied to the

current population, a new population will be formed

and the generational counter will be increased by

one. This process will contin

This method can be viewed as a simple genetic

algorithm that uses the LMI tool to find the fitness

of the resulting candidate individuals of the problem,

under consideration.

ber of generations is attained or some form of

convergence criterion is met.

A s

To be more specific, consider the modified

genetic algorithm depicted in Fig. 2. This algorithm

is similar to the simple genetic algorithm of Fig.1,

with the difference that an LMI optimization

mechanism is used, to compute the objective value

of the current candidate solutions.

AN LMI-BASED GENETIC ALGORITHM FOR GUARANTEED COST CONTROL

329

Genetic

Algorithm

LMI

Optimization

S, T, δ

Fitness

Assignment

Crossover

Selection

Mutation

Reinsertion

Population

Figure 2: The proposed LMI-based Genetic Algorithm

The LMI optimization procedure described in

Fig.2, consists of the following Theorem which is a

modified version of Theorem 3 in (Fischman, 1996)

and stated as,

Theorem

The minimization of the value of the guaranteed

cost (5) for the uncertain system (1) is ensured by a

control law of the form (6), if the optimization

problem described by equation (9) has a non empty

set of feasible solutions (M, W) with M, W being

symmetric positive definite matrices, and P = W

-1

The difference of the above Theorem from

Theorem 3 of (Fischman, 1996) is that the matrices

S, T and the scalar δ, are not variables of the LMI

optimization process, but constant values that come

from the genetic algorithm.

In other words, the genetic procedure manages to

find candidate sets of the (S, T, δ) variables, and

passes these sets into the LMI optimization for

finding optimal value of the matrix P.

The resulted matrix P is used in equation (7) to

derive the corresponding guaranteed cost, with a

known initial condition x

Before the beginning of the proposed algorithm,

one has to take some decisions about the parameters

that must be defined, in order to initialize the

procedure. Some of these parameters are (Coley,

2001)

• Type of individual representation (real,

binary, etc.)

• Population size (typical values are 20-

100)

• Length of individuals (L, depends on

the range of the parameters)

• Crossover probability (P

: typical

values are 0.4-0.9)

• Mutation probability (P

=1/L, 0.01)

• The selection operator

• Number of Generations

The proposed algorithm can be summarized in the

following steps:

Algorithm

Step 1: Generation of the initial population,

consisting of 100 individuals. Each

individual contains, three variables (S,

T, δ), in binary representation, of

length 20 each one.

Step 2: The candidate sets of the variables (S,

T, δ) are appeared as solutions of the

problem. Each one of this set, is

passed as constant matrices into

equation (9), and the LMI procedure is

started.

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

330

Step 3: The resulted matrix P is used to

compute the guaranteed cost that

corresponds to the set of (S, T, δ), by

using (7).

⎥

⎦

⎤

⎢

⎣

⎡

−

−−

00000.100

3229.17070.02855.01002.0

0208.40024.00100.10482.0

4555.00188.00271.00366.0

()

⎪

⎩

⎪

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

−−+−−

⎥

⎦

⎤

⎢

⎣

⎡

−

−−

min

TBGR

SEW

WGBRWEFTFDSDBBRAWWA

MTr

TTTTTT

δδ

(9)

Step 4: The computed cost J, consists the

objective value of the respective set

(S, T, δ), and is used for the fitness

assignment operator of the genetic

algorithm.

⎥

⎦

⎤

⎢

⎣

⎡

−

9900.45200.5

5922.70447.3

1761.04422.0

Step 5: The algorithm continues, by applying

the genetic operators, presented in

Fig.2, with the appropriate settings.

with the uncertainty matrices:

Step 6: A new population is obtained and the

algorithm goes to Step 2 until a fixed

number of generations is achieved.

()

() ()

⎥

⎦

⎤

⎢

⎣

⎡

=∆

0000

2031.102192.00

0000

tata

When the above algorithm has been terminated,

the individual with the higher fitness value is the

solution of the respective problem. Thus the

variables that consist the fitter individual, is the

optimum set of problem variables (S, T, δ), which

obtains a minimum guaranteed cost control J*.

()

⎥

⎦

⎤

⎢

⎣

⎡

=∆

00673.1

The above algorithm is now illustrated by means

of a numerical example.

Let the uncertainty decomposition matrices be as

follows:

5 NUMERICAL EXAMPLE

[]

,101822.00,

2031.1

⎥

⎦

⎤

⎢

⎣

⎡

= ED

A common problem (Kosmidou, 1996, Fischman,

1996) is considered in this section in order to

investigate the performance that can be achieved by

using the proposed method.

This problem represents a 4

order model of a

helicopter in a vertical plane for an airspeed range of

60 knots to 170 knots. For this range of operating

conditions significant changes occur in elements a

and b

of the system matrices, where the

nominal matrices are the following:

[]

05.0,

1346.2

⎥

⎦

⎤

⎢

⎣

⎡

= GF

while the R

= I

and Q

= I

AN LMI-BASED GENETIC ALGORITHM FOR GUARANTEED COST CONTROL

331

(a)

(b) (c)

Figure 3: Fitness landscape in three cross sections, of the proposed genetic algorithm, (a) ST, (b) SZ and (c) TZ cross sections

(a)

(b)

(c)

Figure 4: The complex plane with the poles of (a) open system, (b) closed loop system with the control law of (

Fischman,

1996),

and (c) closed loop system with the control law obtained by the proposed algorithm

By applying the proposed LMI-based Genetic

algorithm the following values of the three degrees

of freedom are obtained

S = 2.1663 T = 1.2036 d = 2.4998

The corresponding guaranteed cost control law

is:

() ()

txtu

⎥

⎦

⎤

⎢

⎣

⎡

∧

1.8963-0.3127-1.59561.2727

3.88762.4866 0.6334 2.5665-

The guaranteed cost obtained by the proposed

method is lower than those of the previous methods

(Kosmidou, 1996, Fischman, 1996), and it is more

close to the optimal one, as illustrated in the

following Table 1. For the computation of the

guaranteed cost, equation (7) is used with initial

condition

[

]

′

= 1111

Table 1: Guaranteed cost and gain norms for the case of

the three methods

Method

Guaranteed

Cost

Norm of the gain

Proposed 4.8978 5.7141

Fischman’s 5.8000 3.0400

Kosmidou’s 5.2591 4.4038

Optimal

Control

3.4890 2.0545

In Figure 3, the landscape of the proposed

genetic algorithm is presented in three cross

sections. The combination of these three landscapes

consists the entire search space of the algorithm.

Also, in Figure 4, the poles of open system and

the closed loop system, for a range of uncertainties

(|a

(t)|

≤

1, |b

(t)|

≤

1) applying the control law derived

by using the proposed method and the method of

(Fischman, 1996) are depicted. As can be seen, by

this figure, the open system is unstable, while in the

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

332

case of the proposed law, is more stable than the one

obtained by (Fischman, 1996).

Finally, the state responses of the closed

systems, in the case of the proposed law and the

(Fischman, 1996) one, are presented in Figure 5.

This figure points, that the proposed gain feedback,

behaves quite more efficiently than the other, since

the responses are smoother.

(a)

(b)

Figure 5: State responses (a) proposed method and (b)

Fischman’s method

6 CONCLUSIONS

A novel method of finding a robust control law, that

guarantees an upper bound of a quadratic

performance index, has been presented in this paper.

The proposed method is based on the

combination of the Genetic Algorithms and the LMI

optimization tool (Gahinet, 1995). It makes use of an

LMI approach to the guaranteed cost, presented in

(Fischman, 1996), of the form of an objective

computation method.

The LMI proposed in (Fischman, 1996), is

modified to find only an optimal matrix P, while the

rest of free parameters are derived through a

genetically processed algorithm.

The results are very promising, since the resulted

guaranteed cost is lower than previous ones

(Kosmidou, 1996, Fischman, 1996),

with an

additional quite better system behavior, in the sense

of stability.

REFERENCES

Coley D.A., 2001. “An Introduction to Genetic

Algorithms for Scientists and Engineers”, World

Scientific Publishing.

Fischman A., Dion J.M., Dugard L. and Trofino Neto A.,

1996. “A Linear Matrix Inequality Approach for

Guaranteed Cost Control”, IFAC World Congress, San

Francisco-USA.

Gahinet P., Nemirovski A., Laub A.J., Chiladi M., 1995.

“LMI Control Toolbox”, The Mathworks Partner

Series.

Holland J.H., 2001. “Adaptation in Natural and Artificial

Systems”, MIT Press.

Kosmidou O.I., Abou-Kandil H., Bertrand P., 1991. “A

Game Theoritic Approach for Guarantedd Cost

Control”, European Control Conference (ECC1991),

pp. 2220-2225, Grenoble-France.

Kundu S., Kawata S., 1996. “Genetic Algorithms for

Optimal Feedback Control Design”, Artif. Intell. Vol.

9, No. 4, pp. 403-411.

AN LMI-BASED GENETIC ALGORITHM FOR GUARANTEED COST CONTROL

333