3.4 Evolutionary scheme of the proposed genetic algorithm

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新埔技術學院改善師資案成果報告書

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內容:
以信號能量為基礎之多目標基因演算法對於間隔系統強健控制器設計之研究

本計劃係探討強健控制器
(r潢ust controller)
之設計法則,以使不確定間隔系統
(i湴erval
system)
得到穩定。主要原理係根據:穩定系統
(stable system)
其脈衝響應能量
(imp畬se
res灯nse

ener杹g

須為有限
(fi湩te)
的觀念,並藉由一公式求得閉迴路系統中與
Kharitonov
多項式相關聯之
4
個頂點系統
(vertex pla湴)
之脈衝響應能量,由於所求得之脈衝響應能量
函數僅對頂點系統的轉移函數係數作處理,因此透過非常簡單之代數運算,即可根據穩定
系統其脈衝響應能量為有限的條件,將強健控制器之參數識別問題轉換成一多目標最佳化
(m畬ti潢jecti癥灴imization)
問題;再將該多目標最佳化問題轉換成一單目標多重不等式限
制條件最佳化
(Mono
-
潢oecti癥灴imization

with m畬tiple i湥quality⁣o湳trai湴s)
問題,並透過
配置有適合度評定機制
(Fitness
assi杮ge湴

mec桡湩sm)
之基因演算法來尋求解答,以求得使
該不確定間隔系統強健穩定之控制器參數;另外,藉由所提出之基因演算法之搜尋,可使
系統對
4
個頂點系統之脈衝響應能量極小化,求得一使該不確定間隔系統得到穩定之非保

(non
-
conservati癥)
的最佳解,並且解除習知技術對於控制器階數上之限制。本計劃最後
將使用既有文獻之範例,以所提出之基因演算法則求得之結果與之
相較,以驗證所提出方
法之可行性。


† †










系科中心

† †









許陳鑑


† †


電子系


† †


助理教授




2

新埔技術學院專題研究



以信號能量為基礎之多目標基因演算法對於間
隔系統強健控制器設計之研究






電子工程系


助理教授


許陳鑑


92

3

6



3

Robust Control of Interval Plants from Signal Energy Point of
View Using Genetic Algorithms


C
hen
-
Chien Hsu
*

and
C
hih
-
Yung Yu
#


*
Department of Electronic Engineering,

St. John's & St. Mary's Institute of Technology,

499 Tam King Rd., Sec. 4, Tam
-
Sui, Taipei, Taiwan, 25135.

E
-
mail:
jameshsu@mail.sjs
mit.edu.tw


#
Department of Computer Science and Information Engineering
,

National Taiwan
University

of Science and Technology
,

43 Keelung Rd
.
, Section 4,

Taipei, Taiwan
,

106.


Keywords:


robust control, interval systems, signal energy, genetic algorithms,

multi
-
objective optimization, stabilizing controllers.


A
BSTRACT

Design of a robust controller which stabilizes an interval plant from the signal energy point
of view via genetic algorithms is proposed in this paper.
When a controller is placed in
series
with the interval plant and closed under unity feedback, it is understood that the
closed
-
loop system can also be characterized as an interval family.
Because stable
system
s
always possess finite impulse response energy, we can obtain the continuous signal

energy
for each of the four closed
-
loop vertex systems associated with the four Kharitonov
polynomials
. With symbolic manipulation of the coefficients of
the

transfer function of the
vertex systems, the parameter identification problem of a robust control
ler can be
transformed into a multi
-
objective optimization problem. A proposed genetic algorithm
incorporating a fitness assignment mechanism

is then used to search for a set of optimal
parameters for the controller which stabilizes the interval plant by m
inimizing the
aggregated continuous signal energy of the four
vertex

systems. The constraints on higher
order plants and controller order commonly encountered by
conventional

design methods
are therefore removed. Several examples are
illustrated

to demonst
rate the effectiveness of
the proposed approach.



4

1. Introduction

Although progress has been made in the realm of robust stability and control of
uncertain systems [1,2], this is still an open problem for which very few results are
available. As pointed o
ut by Bhattacharyya [2] that a significant
deficiency

of control
theory at the present time is the lack of non
-
conservative design methods to achieve
robustness under parameter uncertainty. Generally speaking, most of the existing results in
the area of pa
rametric robust control are analysis results [3].

As an early
attempt

to fill the gap between analysis and design results, an

methods [4] is proposed to synthesize a robust stabilizer for interval plants, in which
structured paramet
ric uncertainties are over
-
bounded by unstructured uncertainties. The
simplicity of this approach, however, contrasts with the inherent conservatism induced by
the uncertainty over
-
bounding. Another design method based on extreme points
results

is
proposed

in [1]. Though worked to some extents, this approach, however, is essentially
graphical and is restricted to two
-
parameter controller design.
There is a parameter plane
method based on a gain phase margin tester and the four Kharitonov vertex polynomials
to
derive a stabilizing controller [
5
]. Once again, this approach is restricted to graphical
presentation with essentially two
-
parameter controller design.

A
synthesis

method of
stabilizing controllers based on analytic
-
real positive (ARP) functions [6] is

proposed,
however, it is restricted to a class of one
-
parameter interval plants only. To make it
possible

to deal with classes of higher
-
order controllers rather than first

order ones, an Alternating
Hurwitz Minor Condition (AHMC) [7] is introduced to enl
arge the class of
polynomial

families for which extreme point results can be obtained. As an overview of recent results
on robust stability and
performance
, a unifying framework is introduced by providing
extremality results for various classes of uncertai
n systems [8]. Based on Hermite
-
Fujiwara
matrices and the
generalized

Kharitonov

s theorem, a sufficient condition is derived for the
existence of a robustly stabilizing
controller

[3]. The condition is then formulated as a
non
-
convex rank
-
one LMI feasibil
ity problem in the controller parameters. While the
above
-
mentioned results are appealing in principle, the restrictive conditions under which
they are derived represent a severe
limitation

from applicability of view [8]. In particular,
there is virtually
no systematic computationally efficient technique of designing a
stabilizing
controller

for high
-
order interval plants [9,24].


5

To circumvent the above
-
mentioned problems and difficulties, soft computing
methods [10
-
13] are emerging to address the design pr
oblem of robust controllers. G
enetic
algorithms

(GA)

[
14,15
]
, in particular, with
their capabilities of directed random search for
global optimization
, are well suited to the design of robust controllers as long as a set of
objective functions can be prope
rly formulated.
Thanks to a probabilistic search procedure
based on the mechanics of natural selection and natural genetics, the genetic algorithms are
highly effective and robust over a broad spectrum of problems [
16
].
These algorithms have
been proven to

be efficient for complex control problems, particularly those involving
optimization of non
-
commensurable and competing multiobjective performance indices
[17,18], which are not computationally tractable using other approaches.

Motivated by suitability e
valuation of digitally redesigned systems from the
signal

energy point of view [19], the design problem of a stabilizing controller for interval plants
is then formulated into the minimization of the signal energy of the vertex systems of the
closed
-
loop s
ystem. As is well known in control literature that a causal system is stable if
and only if its associated impulse response energy is finite. Also, stability of an interval
system can be assured as long as the four Kharitonov polynomials are stable [20]. T
herefore,
robustness can be guaranteed if the
impulse

response energy for each of the four vertex
systems associated with the four Kharitonov polynomials is finite. By doing so, the
problem of designing a robust controller for an interval system is formula
ted as a
multi
-
objective optimization problem (MOP) of the associated signal energy of the impulse
response of the four vertex systems. It is therefore the objective of this paper to

reformulate
the multi
-
objective problem in designing a robust controller
into a
mono
-
objective
optimization problem with multiple inequality constraints by minimizing the aggregated
impulse response energy of the four
vertex

systems. Due to the non
-
convex nature of the
objective function, a proposed GA
-
based approach incorporat
ing a fitness evaluation
mechanism is used to obtain a set of optimal parameters of the controller stabilizing the
interval plant. The constraints on higher
-
order plants and controller order commonly
encountered by
conventional

design methods are therefore

removed by using the proposed
GA
-
based approach.

The paper is organized as
follows
.
Section 2 states the
problem formulation of robust
stabilization of an interval plant
.
Section 3 formulates the problem to
identify

the robust

6

controller into a multiobjec
tive optimization problem, which is subsequently solved by the
proposed genetic algorithm.
Several examples are illustrated in Section
4

to show the
effectiveness of this approach. Conclusions are drawn in Section
5
.


2.

Problem formulation of robust stab
ilization of an interval plant

An interval plant provides a simple and general way to model parametric uncertainty and is
described by a ratio of interval polynomials [3]










(
1
)

where






(
2
)







(
3
)

and the coefficients of
polynomials

a
(s) and
b
(s) vary
independently

in given intervals, i.e.,


,
i
=0, 1, 2, 3,


.



(
4
)

We consider a controller










(
5
)

where






(
6
)


.




(
7
)



Fig. 1 Robust controller
C
(s) for interval plant
G
(s)


When controller
C
(s) is placed in se
ries with plant
G
(s) and closed under unity feedback as
shown in Fig. 1, the transfer function of the closed
-
loop system becomes:


7






(
8
)


Now the problem of robust stabilization of an interval plant can be
formulated

a
s:
Given
interval polynomials a(s) and b(s), find polynomials p(s) and q(s) such that interval
polynomial

D(s) is stable.



Vertex systems of the closed
-
loop system

Clearly, the characteristic
polynomial

of the closed
-
loop system as shown in Fig. 1, i.e.,






(
9
)

is also an interval
polynomial
, where
d
i

are the characteristic coefficients, and

,
i
=1, 2, 3,







(
10
)

According to the Kharitonov

s theorem [20], every interval
polynom
ial

in the family
D
(s) is
Hurwitz if and only if the following four Kharitonov polynomials are Hurwitz,





(
11
)





(
12
)





(
13
)

.




(
14
)

It follows that the entire family is Hurwitz stable if and only if the four vertex
polynomials

D
1
(s)~
D
4
(s) are all Hurwitz stable. This is
equivalently

true that four vertex systems
G
i
(s)
associated with the four Kharitonov pol
ynomials
D
1
(s)~
D
4
(s)







(
15
)

need to be Hurwitz to guarantee the stability of the entire interval family. Instead of dealing
with the vertex
polynomials

D
1
(s)~
D
4
(s), we focus on the vertex systems
G
i
(s) of the
closed
-
loop system to investigate the parameter identification problem of the robust
controller from the signal energy point of view. From the computation point of view,
excessive number of evaluations of robust stability test of 16
-
plant or 32
-
segment result is

8

also

prevented.


3. Identification of the controller parameters via genetic algorithms

We say that a casual system is externally stable if a bounded input,
, produces a bounded output
. A
well
-
known necessary
and sufficient condition for such BIBO (bounded
-
input
bounded
-
output) stability is that the impulse
response

h
(
t
) be such that









(
16
)

The requirement that the
absolute

value of the impulse response
h
(t), integrated

over an
infinite

range, is finite implies that the signal energy of the impulse response
h
(
t
) needs to
be finite as well to guarantee stability of the system. That is,









(
17
)

To guarantee the stability of the
ver
tex systems
G
i
(s) in Eq. (15), their associated
continuous signal energy (CSE)

f
i
(
p
j
,q
j
) needs to be finite. That is,


,
i
={1,2,3,4}



(
18
)

where









(
19
)

is the impulse response

of the vertex system
G
i
(s), and
Q
i

is a hypothetical impulse
response energy of finite magnitude for each vertex system
G
i
(s).

To evaluate the continuous signal energy in Eq.(18), however, we need to symbolically
manipulate the integration, which is gen
erally
impractical

if not possible. Fortunately,
there
is a
n

elegant

closed
-
form formula developed in [
21
] of


,
i
={1,2,3,4}

(
20
)

where
n

is the order of
the closed
-
loop vertex systems
G
i
(s),

and

l
,

l


are the coeffici
ents
of the Alpha and Beta tables of
the vertex systems
G
i
(s)

[
21,22
].

Note that

l

and


l

are
functions of the controller parameters
p
j

and
q
j

to be identified.

To this far, the robust controller design problem for an interval plant has been

9

formulated as

a multi
-
objective optimization problem as shown in Eq.(18). However, the
continuous signal energy in Eq. (20) is a non
-
convex function in general in the searching
space of the controller parameter domain. Gradient
-
based optimization algorithms generally
l
ead to solutions that have local properties only.
Genetic algorithms, with their power as an
efficient and robust alternative for solving complex and highly nonlinear optimization
problem, will be used to identify parameters of the robust controller.


3.1

Preliminaries of the genetic algorithms

Basically, genetic algorithms are probabilistic algorithms which maintain a population of
individuals (chromosomes, vectors),
, for iteration
t
. Each
chromosome

represe
nts a potential solution to the problem at hand, and is evaluated to
give some measure of its “fitness”. Then, a new population is formed by selecting the more
fit individuals. Some members of the new population undergo transformations by means of
genetic
operators to form new solutions. After some number of generations, it is hoped that
the system converges with a near
-
optimal solution [
14
].

There are two primary groups of genetic operators,
crossover

and
mutation
, used by
most researchers. Crossover comb
ines the features of two parent chromosomes to form two
similar offspring by swapping corresponding segments of the parents. The intuition behind
the applicability of the crossover operator is information exchange between potential
solutions. Mutation, on
the other hand, arbitrarily alters one or more genes of a selected
chromosome, by a random change with a probability equal to the mutation rate. The
intuition behind the mutation operator is the introduction of some extra variability into the
population.
V
arious variations of the operator
s

are developed to perform different
responsibilities like fine tuning and prevention of premature convergence [
14
]

of the
genetic algorithms
.


3.2 Mono
-
objective genetic algorithm with multiple inequality
constraints

With
the multiobjective problem presented in Eq. (18), the simplest and
natural

way is to
have it reformulated as a mono
-
objective optimization problem with multiple
constraints

by
means of an aggregation function. This means that the multiobjective
optimizatio
n

problem
of Eq. (18) is transformed into a scalar optimization of the weighted continuous signal

10

energy with 4 inequality constraints of the form

Minimize








(
21
)

Subject to










(
22
)





(
23
)

where

are the weighting coefficients representing the relative importance of the
objective functions [14,23]. Combining the objectives to obtain an optimized solution has

the advantage of producing a single solution, because the design problem at hand requires
no
interaction

with the decision making

among the controllers derived by the proposed
approach

as far as stabilizing controllers are concerned.


In order to handle m
ultiple inequality constraints in GA as demonstrated in Eqs.(21
-
23),
we use a method based on penalty function and tournament selection, where two solutions
are compared at a time and the following criteria are always enforced [23].

1.

Any feasible solution
is preferred to any infeasible solution.

2.

Among two feasible solutions, the one having better objective function value is
preferred.

3.

Among two
infeasible

solutions, the one have smaller constraint violation is
preferred.


3.3 Fitness evaluation mechanism

Th
e performance of each chromosome is evaluated according to it fitness. After generations
of evolution, it is expected that the genetic algorithms converges and a best chromosome
with largest fitness representing the optimal solution to the problem is obtai
ned. As shown
in Eq.(15), there are 4 vertex systems
G
j
(s),
j
=1~4, corresponding to every chromosome
X
i

in a population
P
(t). For
every stable vertex system
G
j
(s)

(
with

finite signal energy
)
corresponding to
chromosome
X
i
, we
calculate the continuous signa
l energy
CSE
j
(
i
), which
is subsequently normalized as:







(
24
)

where
is the continuous signal energy of the
j
th vertex system

11

G
j
(s),
j
=1~4, and

is the maximum continuous
signal energy of
the vertex system
G
j
(s),
j
=1~4, for all chromosome
X
i

in the
current

population. For every
unstable
vertex system
G
j
(s)

(with
no
finite signal energy)

corresponding to
chromosome
X
i
,
let
NCSE
j
(
i
)=4.

To this far, we have established a norm
alized continuous signal energy
NCSE
j
(
i
) for
the
j
th vertex system
G
j
(s),
j
=1~4, corresponding to the
i
th chromosome
X
i

in the current
population
P
(t). It is clear that
NCSE
j
(
i
) has a value of either 0~0.9 (stable Kharitonov
Polynomial
s), or 4 (unstable Kh
aritonov
Polynomial
s).


Now the fitness function
Fitness
(
X
i
) can be defined as:


(
25
)

where








(
26
)

and
E
small_unstable

is the smallest signal energy for all
E(i)

corresponding to

an infeasible
chromosome
X
i
. By doing so, we have devised an evaluation
mechanism

which ranges the
fitness distribution of Eq.(25) into 5 groupings listed in Table 1 below.


Table 1 Fitness Grouping of
Fitness
(
X
i
)

No. of stable
Kharitonov
p
olynomials

E(i
)

Fitness range of
Fitness
(
X
i
)

0

4
×
4

17
-

4
×
4 = 1

1

4
×
3 + (0~0.9)

17
-

[4
×
3 + (0~0.9)] = (4.1~5)

2

4
×
2 + (0~0.9)
×
2

17
-

[4
×
2 + (0~0.9)
×
2] = (5.2~7)

3

4 + (0~0.9)
×
3

17
-

[4 + (0~0.9)
×
3] = (10.3~13)

4

(0~0.9)
×
4

17

(ま〮㤩
×
㐠⬠ㄷ4
-

(まㄶ1‽
ㄴ⸴1㌴P


ft is clear from Table ㄠ that chromosomes res畬ti湧 i渠 m潲e stable hharitonov
p
潬祮omials

潲 smaller si杮al
ener杹

灯psess larger fitness val略Ⱐan搠癩ce 癥rsa⸠Base搠on
t桩s fit湥ss evaluation mecha湩smⰠ the pr潰潳e搠 dA can efficientl礠 evol癥 toward an

timal s潬uti潮 t漠潢oain the sta扩lizi湧 c潮tr潬ler.


12


3.4 Evolutionary scheme of the proposed genetic algorithm

Evolutionary
scheme
of the proposed mono
-
objective genetic algorithm with multiple
inequality
constraints

includes the steps of population initi
alization and r
eproduction
o
peration

based on tournament selection. Three genetic operators: w
hole
a
rithmetic
c
rossover
, heuristic c
rossover
, and uniform
mutation

are performed on the selected
chromosomes after the reproduction operation to
keep the balanc
e between the population
diversity and selective pressure during the evolution process.

To prevent the loss of the
optimal solution ever searched and increase the convergence
rate
, the elitist replacement
is

adopt
ed

to preserve the optimal solution in the
current generation.


3
.
5

Computational Algorithms

The proposed
approach to derive the r
obust control
ler

of uncertain interval systems

via
genetic algorithm
is supplemented by a computational algorithm below, which can be easily
implemented using Matlab.

St
ep
1
:
(Preparation)

Specify
coefficient parameters of the interval plant
G
(s) and controller
C
(
s
), and

genetic algorithms parameters: population size (
pop_size
), maximum generation
(
max_gen
), crossover rate (
p
c
), mutation rate (
p
m
), and

tournament size

(
k
)
.

Step
2
:
(Initialization)

1)

Set the best solution
X
*
=0, best fitness value
f
max
= 0
, and generation number
t
=1
.


2)

Generate a
n

initial population
P
(
t
)

of
pop_size

chromosomes within
X
i
=
[
lower_bound, upper_bound
]
,

for
i
=1 to

pop_size
.

Step
3
:
.

(Evaluation)

1)

For

every stable vertex system
G
j
(s),
j
=1~4, corresponding to chromosome
X
i
,
i
=1~
pop_size
,

1.1)

Calculate the continuous signal energy
CSE
j
(i).

1.2)

Find

1.3)

Calculate the normalized
continuous signal energy


13


2)

For every unsta
ble vertex system
G
j
(s),
j
=1~4, corresponding to
chromosome
X
i
,
i
=1~
pop_size
, let
.

3)

For each chromosome
X
i
,
calculate

.

4)

Find the smallest signal energy
E
smallest_unstable

for all
E(i)

corresponding to an
infeas
ible chromosome
X
i
.

5)

Calculate
Fitness
(
X
i
) as:


6)

If
F
itness
(X
i
)> f
max
, then
X
*
=
X
i

and
f
max
=

F
itness
(X
i
)
for

i
=1 to

pop_size.

Step
5
: (Reproduction)

1)

Randomly select some number
k

of chromosomes and selects the best one
from this set o
f
k

elements into the next generation.

2)

Repeat
pop_size

times

Step
6
: (Crossover)

1)

Randomly s
elect
n

(even) chromosomes

X
i
,
for
i
=1 to
n
, in population
P(t)

according the crossover rate
p
c
, and then perform the arithmetic crossover
and heuristic crossover op
eration to produce offspring

Y
i
, for

i
=1 to
n
.

2)

Set

X
i
=

Y
i
, for
i
=1 to
n
.

Step
8
: (
U
niform Mutation)

1)

Randomly s
elect
n

chromosomes

X
i
,

for
i
=1 to
n
,

in population
P(t)

according the mutation rate
p
m
, and then perform the uniform mutation to
produce the mut
ated genes

Y
i
.

2)


Set

X
i
=
Y
i
.

Step
9
: (Elitist replacement)

If
F
itness
(X
i
)< f
max
,

for all
i
=1 to
pop_size
,

then
the solution

with the
smallest fitness value in the current
generation

is replaced by the best solution

X
*
.

Step
10


14

If
t
=
m
ax_gen
then output best solution
X
*

with a best fitness value of

f
max
; else
t=t+1

and goto Step

3.


4
. Illustrated Examples

Example

1

Consider the interval plant [5]


where the plant parameters vary as follows
:


A PID controller, which has the form of


is used to stabilize the interval plant, where
K
D
,
K
P
, and
K
I

are parameters to be identified.

The result given in Fig. 2 illustrates the evolution process to search

for a robust
controller by the proposed genetic algorithm. Note that the max fitness function is shown in
solid line, while a dashed line gives the largest fitness value of all infeasible chromosomes
in the current generation. The generation number at
whi
ch

the solid line deviates from the
dashed line is called a Deviation Point (DP). The next generation after the DP provides the
first answer of a robust controller, although the signal energy is not minimized. With
reference

to Fig. 2, the deviation point
(DP) appears at generation 4. That is, a robust
controller can be first found at generation 5. When the GA program (pc=1, pm=0.4, and
population

size of 20) terminates at generation 20, a robust controller is obtained as:



For verif
ication purpose, the closed
-
loop system
incorporating

the controller
C
(s) is
listed below:

, which is apparently an interval system itself. Four
Kharitonov
polynomials

associated
with
D
(s) are:


15


, each corresponding to a
CSE

for
G
i
(s)=1/
D
i
(s),
i
=1,2,3,4.

For example,


where the Alpha and Beta tables are listed below for easy reference.

α
1

0.285714

β
1

0

α
2

0.350808

β
2

0

α
3

0.091740

β
3

0

α
4

0.673818

β
4

0

α
5

1.074163

β
5

0.000000665539


Similarly,
CSE
2
,
CSE
3
, and
CSE
4

can be obtained in the same way as
0.204846×
10
-
12
,
0.208495×
10
-
12
, and
0.205399×
10
-
12
,
respectively
. In light of the fin
ite signal energy
CSE

presented for the vertex systems
G
i
(s), the four associated
Kharitonov
polynomials

are
Hurwitz stable, and hence the closed
-
loop system
G
cl
(s) is stable.

Fig. 3 shows the fitness distribution as the function of the controller paramete
rs
K
P

and
K
I
. Note that there are basically 5 groups of fitness in the perspective view in Fig. 3,
representing 0, 1, 2, 3, and 4 stable
Kharitonov
polynomials

for the corresponding
parameter set,
respectively
. Parameter sets having fewer stable
Kharitonov

polynomials

possess smaller fitness, and vice versa. That is, chromosomes closer to the stable area result
a higher fitness, while chromosomes far away from the stable area possess less fitness,
which is a perfect match to the evaluation mechanism derived

in Eq.(25) in terms of
evolving direction toward a better solution.

Note that the stabilizing controller is obtained

b
y dealing with only four continuous
signal energies of
the

four vertex
systems
, thus
pre
v
ent
ing the excessive
number of
evaluations of ro
bust stability test of 16
-
plant or 32
-
segment results [12]
. Also, the
derivation process of proposed approach requires no
graphical interpretation as compared
with several existing methods [1,5]. In general, the stability controller can be efficiently
obta
ined within a moderate number of iterations by
using

the proposed approach without
suffering

from the inherent shortcomings.


16


Fig. 2 Fitness function
Fitness
(
X
) in Example 1.



Fig. 3 Perspective view of the fitness distribution of Example 1

when the PID controller
parameter
K
D

is fixed at 0.01.

Stable Area

High

Low

Fitness Scale

Deviation Pt.

K
P

K
I

Max fitness

of all chromosomes

Largest fitness

o
f infeasible chromosomes


17

Example

2


Consider a higher order interval plant [
9
] given by


A second order controller given by



with the bounds on

are relaxed as


as compared to those of Example 2 in [9].

With

reference to Fig. 4, the DP
appears

at generation 11, which implies a
stabilizing

controller can be first obtained at generation 12. When the GA program ter
minates at
generation 20, a robust controller
C
(s,
q
) stabilizing the interval plant is evolutionarily
obtained

by the proposed GA
-
based approach as:


with a population size of 100,
p
c
=1, and
p
m
=0.4.


For verification purpose, the fo
ur
Kharitonov
polynomials

associated with the
closed
-
loop system are listed below:


, each corresponding to a
CSE

of 1.0246
×
10
-
39
, 9.2795
×
10
-
40
, 1.2919
×
10
-
39
, and

18

1.2207
×
10
-
39
,
respectively
, for
G
i
(s)=1/
D
i
(s),
i
=1,2,3,4. In light of
the finite signal energy
presented for the vertex systems
G
i
(s), the four associated
Kharitonov
polynomials

are
Hurwitz stable, and hence the closed
-
loop system
G
cl
(s) is stable.



Fig. 4 Fitness function
Fitness
(
X
) in Example 2.

As s
hown in this example, the interval plant has a high order of 5 and the controller
C
(s) is of 2
nd

order (instead of 1
st

order) consisting of 6 parameters to be identified, which
creates a virtually impractical, if not impossible, calculating burden for exis
ting techniques
to compute a such stabilizing controller [7,9]. However,
the

proposed GA can obtain a
desired solution with little effort.


Example

3

Consider the interval plant [2,3]


where
ε

0 parametrizes the size of the uncerta
inty intervals.

The robust first
-
order controller stabilizing the interval plant is evolutionarily obtained
Deviation Pt.


19

via the proposed genetic algorithm
using a population
size

of
100
, pc=0.8, and pm=0.1. As
demonstrated

in Fig. 5, the deviation point (DP) appears
at generation 24, and a solution for
the robust controller is first found at generation 25. When the GA terminates at generation
50, we obtain the robust controller of








(
27
)


Fig. 5 Fitness fu
nction
Fitness
(
X
) in Example 3.


Four
Kharitonov
polynomials

associated
with the

characteristic polynomial are:


, each corresponding to a finite
CSE

for
G
i
(s)=1/
D
i
(s),
i
=1,2,3,4. Therefore, the closed
-
loop
system is stable.


Note th
at the uncertainty interval size
ε

allowed, which characterizes the
performance

margin of the derived controller, of the stabilizing controller developed in [2]
Deviation Pt.


20

is restricted to
ε

6.321, and the stabilizing controller proposed in [3] using the Frank and
Wol
fe algorithm has a restriction of
ε

7.5. No stabilizing controller can be found
beyond that limit
because

of the local opt
imum

reached. On the other hand, the robust
controller
C
(s) in Eq. (27) obtained by using the proposed method has a desired margin of
ε

8, which offers a significant relaxation on the restriction over the existing methods.


Example

4


Consider the interval plant [2,3]


where
ε

0 parametrizes the size of the uncertainty intervals.

The robust controller stabilizing

the interval plant is evolutionarily obtained via the
proposed genetic algorithm using a population
size

of
100
, pc=0.8, and pm=0.1 as:








(
28
)


Note that the uncertainty interval size
ε

allowed for the stabilizing
controller
obtained in [2] is restricted to
ε

4.
Although

the results revealed in [3] using the Potential
reduction algorithm are improved and a stabilizing controller is found for
ε

6.8. No
further improvement beyond that limit is possible. On the other ha
nd, the robust controller
C
(s) in Eq. (28) obtained by using the proposed method further relaxes the restriction to
ε

6.9 without much effort (12
generations

only).


5.
Conclusions


In this paper,
the problem of designing a stabilizing controller for an int
erval plant is
successfully transformed into a multiobjective optimization problem from the signal energy
point of view. A proposed genetic algorithm incorporating a
suitable

fitness
evaluation

mechanism which minimizes the aggregated function of continuou
s signal energies
associated with the four
Kharitonov
polynomials

is then implemented with the help of
symbolic manipulation tool to evolutionarily derive the robust controller.
Stabilizing
controllers are obtained

b
y dealing with only four continuous sign
al energies associated

21

with the vertex
systems
, thus
pre
v
ent
ing the excessive
number of evaluations of robust
stability test of 16
-
plant or 32
-
segment results
. Also, the derivation process of proposed
approach requires no
graphical interpretation as compar
ed with several existing methods.
Conventional

design
constraints

on the higher
-
order interval plants and controller order are
removed, because there is no restrictive condition under which the proposed approach is
developed. In general, the stability cont
roller can be efficiently obtained within a moderate
number of iterations by
using

the proposed approach without
suffering

from the inherent
shortcomings. Simulation results have demonstrated that the proposed genetic algorithm
with a suitable fitness eval
uation mechanism
efficiently

evolves toward a better solution to
search for a robust controller. Several illustrated examples, including those with
higher
-
order interval plants and
arbitrarily

assigned controller order, have demonstrated the
effectiveness
of the proposed approach. As highlighted by Bhattacharyya that

so little is
known about the solution that every step is of interest

, it is hoped that this paper can lead
to some
preliminary

results toward a practical and effective way in
robust

controlle
r design
for SISO interval plants.



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