A Proposal of Multiobjective Fuzzy Regulator Design for State Space
Nonlinear Systems
Rafael J. M. Santos and Ginalber L. O. Serra
Federal Institute of Education, Science and Technology,
Av. Getúlio Vargas 04, Monte Castelo, CEP: 65030-005, São Luís, Maranhão, Brazil
Keywords:
Fuzzy Regulator, Optimal Control, Linear Quadratic Regulator, Pole Placement, Multiobjective Feasible
Region.
Abstract:
This paper proposes a Takagi-Sugeno (TS) fuzzy regulator design methodology for nonlinear dynamic sys-
tems. The Linear Quadratic Regulator (LQR) and Pole Placement (PP) techniques are combined in a TS fuzzy
structure in order to guarantee an optimal controller with satisfactory transient response based on poles allo-
cated properly. The definition and analysis of the multiobjective feasible region, considering the influence of
the desired poles on the weighting matrices Q and R in the quadratic cost function, are presented. Lyapunov
based stability analysis and simulations results on fuzzy regulator design for a robotic manipulator illustrates
the efficience of the proposed methodology.
1 INTRODUCTION
The actual control problems have natural tendency to
increase its complexity due to the multiobjective per-
formance criterion and its satisfaction with high pre-
cision and accuracy. In this context, the classical con-
trol theory, characterized by the input-output repre-
sentation has limitations because it does not allow full
access to all variables of the plant to be controlled,
which has motivated the search for new control strate-
gies, exploring the state space representation, charac-
terizing the modern control theory. Since then, several
methods have been proposed to deal with linear plants
(in this case, the nonlinearities and uncertainties are
considered negligible) as well as to develop more ad-
vanced controllers taking into account the nonlineari-
ties, uncertainties and time varying parameters of the
plant (Choi, 2007; Li and Tsai, 2007; Chen et al.,
2007a; Zhu and Ma, 2006; Ghosh et al., 2009; Nie
and Tan, 2011; Zhao et al., 2004).
The search for new model based control strate-
gies from computational intelligence techniques, con-
sidering the impact of fuzzy systems, among oth-
ers, has allowed succesfull applications in modeling
and control of complex plants with promising re-
sults (Chen et al., 2007b; Mohammadian et al., 2003;
Pedrycz and Gomide, 2007; Liu et al., 2011; Mum-
ford and Jain, 2009; Gorji and Menhaj., 2008; Eber-
hart and Shi, 2007; Gui-juan et al., 2010; Ko and
Jatskevich, 2007; Luo et al., 2006). The fuzzy sys-
tems theory, in particular, has received great atten-
tion from researchers in the this area to deal effec-
tively with uncertainties and nonlinearities through its
functional structure (Babuška, 1998; Khanesar et al.,
2011; Boulkrounea et al., 2010; Nie and Tan, 2011;
Preitl et al., 2008; Abbas et al., 2011; Jiang et al.,
2008). This research interest has grown in recent
years by the possibility of incorporating in the fuzzy
inference structure the expert knowledge as well as
the mathematical formalism from the modern con-
trol theory, resulting in fuzzy control systems with
high degree of transparency, interpretation, robust-
ness and stability (Eltamaly et al., 2010; Yuana et al.,
2010; Liu et al., 2010; Berrios et al., 2011; Li, 2010;
Ko and Jatskevich, 2007; Torres-Pinzón and Leyva,
2009; Gheysari and Mashoufi, 2011; Shi et al., 2011).
In (Márquez et al., 2009), a general methodology that
uses fuzzy logic to systematically and formally syn-
thesize stable nonlinear control systems design is pro-
posed. Although this methodology is based on Lya-
punov theory, it avoids searching for Lyapunov func-
tions. This allows the synthesis procedure to be sys-
tematic as well as formal and, especially, independent
of heuristics. In (Zhao et al., 2009), a novel robust
fuzzy controller design method is proposed to stabi-
lize a class of chaotic (hyperchaotic) systems with un-
certain parameters based on their equivalent TS fuzzy
models. In this method, the interval system theory is
50
J. M. Santos R. and L. O. Serra G..
A Proposal of Multiobjective Fuzzy Regulator Design for State Space Nonlinear Systems.
DOI: 10.5220/0004018300500058
In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 50-58
ISBN: 978-989-8565-21-1
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
applied to deal with the parametric uncertainty firstly,
and then a fuzzy state feedback controller is designed
to stabilize the equilibrium of the uncertain chaotic
(hyperchaotic) systems robustly based on Exact Lin-
earization (EL) theory and Parallel Distributed Com-
pensation (PDC) technique. The designed controller
with simple structure and rapid response can stabi-
lize many types of uncertain chaotic or hyperchaotic
systems. In (Zhao et al., 2004), after a study on syn-
thesis of neural network and fuzzy logic based con-
trollers for optimally controlling uncertain nonlinear
systems linear in control, three different types of hier-
archical controller architectures are proposed, which
include a hierarchical neuro-fuzzy controller architec-
ture, a hierarchical fuzzy-neuro controller architec-
ture and a hierarchical fuzzy logic controller archi-
tecture. The study concludes the proposed neural-
network and fuzzy logic based control schemes are
useful for nonlinear system applications.
In this paper is proposed a Takagi-Sugeno (TS)
fuzzy state space control strategy for nonlinear dy-
namic systems based on two methodologies: Pole
Placement and Linear Quadratic Regulator. The pole
placement technique is used to design the state feed-
back gain matrix which determines the transient re-
sponse characteristics by allocating the desired closed
loop poles. The Linear Quadratic Regulator, in turn,
is used to compute the state feedback gain matrix con-
sidering the optimization of a quadratic cost func-
tion. The proposal in this paper is to combine these
methodologies, in a fuzzy context, in order to design
an optimal regulator for nonlinear plants with desired
transient response performance. This paper is struc-
tured as follow: Section 2 presents the fundamen-
tal structure of the proposed methodology. Section 3
presents the main computational results of the fuzzy
multiobjective regulator design for a robotic manip-
ulator according to multiobjective feasible region de-
fined by LQR and pole placement techniques. Section
4 presents conclusions and proposals for future work.
2 POLE PLACEMENT BASED
LINEAR QUADRATIC FUZZY
REGULATOR
This section presents the analitical formulation of the
multiobjective fuzzy regulator design from the fusion
of pole placement and LQR methods to garantee a de-
sired transient response with optimal control action
for nonlinear systems.
2.1 State Space Takagi-Sugeno Fuzzy
Modeling
2.1.1 State Space Takagi-Sugeno Fuzzy
Inference System
The Takagi-Sugeno fuzzy inference system uses in
the consequent proposition of its IF-THEN rule base a
functional expression of the linguistic variables in the
antecedent proposition. The i|
[i=1,2,...,l]
-th rule, where
l is the number of rules, is given by
R
i
: IF ˜x
1
is F
i
j| ˜x
1
AND ... AND ˜x
n
is F
i
j| ˜x
n
THEN
˙
x(t) = A
i
x(t) + B
i
u(t)
(1)
The vector
˜
x ∈ R
n
contains the linguistic variables
of the antecedent proposition. Each variable has its
own linguistic universe of discourse U
˜x
1
,...,U
˜x
n
parti-
tioned by fuzzy sets representing the corresponding
linguistic terms. The variable ˜x
t
|
t=1,2,...,n
belongs to
the fuzzy set F
i
j| ˜x
t
with a value γ
i
F
j| ˜x
t
defined by a
membership function γ
i
˜x
t
: R → [0, 1], with γ
i
F
j| ˜x
1
∈
γ
i
F
1| ˜x
1
,γ
i
F
2| ˜x
1
,...,γ
i
F
p
˜x
t
| ˜x
1
, where p
˜x
t
is the number of
partitions of the universe of discourse associated with
the linguistic variable ˜x
t
. The matrices A
i
∈ R
n×n
and
B
i
∈ R
n×1
represent the parameters of the i-th local
state space model of the nonlinear plant on its i-th op-
erating point; x(t) ∈ R
n×1
, is the state vector of the
plant and u(t) ∈ R, is the input vector of the plant.
The fulfillment degree h
i
for the rule i is given by the
t-norm operator:
h
i
= γ
i
F
j| ˜x
∗
1
∧ γ
i
F
j| ˜x
∗
2
∧ ... ∧γ
i
F
j| ˜x
∗
n
(2)
where ˜x
∗
t
is any point in U
˜x
∗
t
. The normalized fulfill-
ment degree for the i-th rule is defined by:
λ
i
(
˜
x) =
h
i
(
˜
x)
l
∑
r=1
h
r
(
˜
x)
(3)
This normalization implies that
l
∑
i=1
λ
i
(
˜
x) = 1 (4)
The TS fuzzy model response is a weighted sum
of the consequents, i.e., a convex combination of the
local state space models:
˙
˜
x(t) =
l
∑
i=1
λ
i
(
˜
x)(A
i
x(t) + B
i
u(t)) (5)
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51
The TS fuzzy model can be considered as a map-
ping from space of the antecedent proposition (input)
to the convex region (polytope) in the space of lo-
cal submodels defined by the functional expressions
in the consequent proposition. This property simpli-
fies the analysis in the context of robust systems for
identification and controllers design (Serra and Bot-
tura, 2009; Bergsten, 2001).
2.2 Fuzzy Regulator Design
Consider a SISO linear plant, representing a local sub-
model for i-th rule, described by
˙
x = A
i
x+ B
i
u (6)
where x ∈ R
n×1
, is the state vector of the plant; u ∈ R,
is the input vector of the plant; A
i
∈ R
n×n
, is the state
matrix; B
i
∈ R
n×1
, is the input matrix.
The quadratic optimal regulator problem consists
in minimizing the objective function given by:
J =
Z
∞
0
x
T
Qx+ u
T
Ru
dt (7)
where Q and R are weighting matrices to be selected
by designer related to the state x vector and control ac-
tion u, respectively. The matrices Q and R must also
be nonnegative definite, which is most easily accom-
plished by picking the Q and R to be diagonal with
all diagonal elements positive or zero. The solution
of this objective function, for the i-th rule, results in a
state feedback optimal control gain matrix K
i
, given
by:
u(t) = −K
i
x(t) (8)
Considering the i-th functional expression in the
consequent proposition of the Takagi-Sugeno fuzzy
inference model, according to equation (1), the closed
loop state feedback fuzzy system is given by:
˙
x = A
i
x− B
i
K
j
u = (A
i
− B
i
K
j
)x (9)
Assuming that the matrix A
i
− B
i
K
j
is stable, it is
possible to obtain a state feedback fuzzy control gain
matrix K
j
from the solution of the algebraic Riccati
equation:
A
T
i
P+ PA
i
− PB
i
R
−1
B
T
i
P+ Q = 0 (10)
for the matrix P. Therefore, the gain K
i
is given by:
K
i
= R
−1
B
T
i
P|
[ j=1,2,...,l]
(11)
Equation (11) provides the following optimal feed-
back fuzzy control action:
u(t) = −K
i
x(t) = −R
−1
B
T
i
Px(t) (12)
Consider, the consequent proposition represented
by a second order functional expression, in the con-
troller canonical form:
˙
x =
0 1
−a
i
2
−a
i
1
x+
0
1
u (13)
The matrices Q and R are defined as
Q =
µ 0
0 1
(14)
R =
β
(15)
where µ ≥ 0 and β > 0.
The matrix P to be obtained, is defined by:
P =
p
11
p
12
p
21
p
22
(16)
where p
12
= p
21
. Substituting the matrix A
i
in (13)
and the matrices Q, R and P defined in (14), (15),
(16), respectively, in equation (10), results
0 -a
i
2
1 -a
i
1
p
11
p
12
p
12
p
22
+
p
11
p
12
p
12
p
22
0 1
-a
i
2
-a
i
1
−
"
p
11
p
12
p
12
p
22
#"
0
1
#
[β]
−1
h
0 1
i
"
p
11
p
12
p
12
p
22
#
+
µ 0
0 1
=
0 0
0 0
(17)
Developing the equation (17), it yields:
−2a
i
2
p
12
−
(p
12
)
2
β
+µ p
11
−a
i
1
p
12
−a
i
2
p
22
−
p
12
p
22
β
p
11
−a
i
1
p
12
−a
i
2
p
22
−
p
12
p
22
β
2p
12
−2a
i
1
p
22
−
(p
22
)
2
β
+1
=
0 0
0 0
(18)
From equation (18), the following equations sys-
tem is obtained:
−2a
i
2
p
12
−
(p
12
)
2
β
+ µ = 0
p
11
− a
i
1
p
12
− a
i
2
p
22
−
p
12
p
22
β
= 0
2p
12
− 2a
1
p
22
−
(p
22
)
2
β
+ 1 = 0
(19)
For the solution of (19), the values of p
11
, p
12
and
p
22
are given by:
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
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p
11
= a
i
1
p
12
+ a
i
2
p
22
+
p
12
p
22
β
p
12
= −a
i
2
β+
q
a
i
2
2
β
2
+ µβ
p
22
= −a
i
1
β+
v
u
u
t
(
a
i
1
)
2
β
2
+2
−a
i
2
β+
r
(
a
i
2
)
2
β
2
+µβ
!
β+β
(20)
From equation (11), it yields:
K
i
=[β]
−1
h
0 1
i
"
p
11
p
12
p
12
p
22
#
=
h
p
12
β
p
22
β
i
(21)
or
K
i
=
k
i
1
k
i
2
(22)
where
k
i
1
= −a
i
2
+
q
(
a
i
2
)
2
β
2
+µβ
β
k
i
2
= −a
i
1
+
s
(
a
i
1
)
2
β
2
+2
−a
i
2
β+
q
(
a
i
2
)
2
β
2
+µβ
β+β
β
(23)
The characteristic equation of the state feedback
system can be determined after obtaining the gain ma-
trix K
j
, as follow:
|sI
2
− A
i
+ B
i
K
j
| =
s
2
+
s
β
(
a
i
1
)
2
β−2a
i
2
β+2
r
β
(
a
i
2
)
2
β+µ
+1
β
s+
+
r
β
(
a
i
2
)
2
β+µ
β
= 0
(24)
Based on the linear quadratic regulator theory and
from (23) and (24), it is possible to assign poles for
each functional expression defined in the i-th rule of
the Takagi-Sugeno fuzzy inference model and design
a respective state feedback gain matrix K
i
ensuring
the optimal conditions defined by matrices Q and R
in terms of µ and β, and desired transient response as
well. In the Appendix, the consistency of the char-
acteristic equation coefficients is shown according to
Caley-Hamilton Theorem.
2.2.1 Multiobjective Feasible Region Analysis
According to the linear quadratic regulator theory and
the equations (23) and (24), it is possible to choose
poles for each submodels and design the correspond-
ing state feedback gain matrix K
i
to garantee the ex-
istence of the matrices Q and R in terms of µ and β.
Consider two generic poles s
1
and s
2
. Assuming the
submodels are causal, the real part of the poles will
be ever negative, in order to ensure stability. Based
on equation (24), it yields:
r
β
a
i
2
2
β+ µ
β
= (s
1
s
2
) (25)
and
a
i
2
2
β+ µ
β
= (s
1
s
2
)
2
And the following relation can be obtained:
µ
β
=
(s
1
s
2
)
2
−
a
i
2
2
(26)
Similarly, it yields:
v
u
u
t
β
(
a
i
1
)
2
β−2a
i
2
β+2
s
β
(
a
i
2
)
2
β+µ
+1
!
β
= −(s
1
+s
2
) (27)
and
(
a
i
1
)
2
β−2a
i
2
β+2
s
β
(
a
i
2
)
2
β+µ
+1
!
β
=(s
1
+s
2
)
2
a
i
1
2
− 2a
i
2
+ 2
r
β
a
i
2
2
β+ µ
β
| {z }
s
1
s
2
+
1
β
= (s
1
+ s
2
)
2
1
β
= s
2
1
+ s
2
2
− 2a
i
2
+
a
i
1
2
β =
1
s
2
1
+ s
2
2
− 2a
i
2
+
a
i
1
2
(28)
Substituting (28) in equation (26):
µ =
(s
1
s
2
)
2
−
a
i
2
2
s
2
1
+ s
2
2
− 2a
i
2
+
a
i
1
2
(29)
Equations (28) and (29) provide a direct relation-
ship between the desired poles to be allocated and the
parameters of the linear submodels in the consequent
proposition of the state space Takagi-Sugeno fuzzy
model. The multiobjective optimal condition, based
on the pole placement method and linear quadratic
regulator problem, slightly restricts the area of the
poles to be chosen adequately, which becomes a sub-
region in the left half of the complex plane, so called
as feasible region. The feasible regions for com-
plex and real poles are shown in Figure 1(a) and
1(b), respectively. As example, the complex poles
s
1
= −0.1 + j0.1 and s
2
= −0.1 − j0.1, implies to
µ = −4 × 10
−4
and β = −1, meaning these poles
AProposalofMultiobjectiveFuzzyRegulatorDesignforStateSpaceNonlinearSystems
53
(a) Complex pole placement
(b) Real pole placement
Figure 1: Multiobjective feasible regions for the pole place-
ment problem.
are out of the feasible region, once the matrices Q
and R from equations (14) and (15) define µ ≥ 0 and
β > 0. The real poles s
1
= −0.5, s
2
= −0.7, implies to
µ = −0.4712 and β = −3.8462, meaning these poles
are inside of the feasible region.
The methodological procedure for the multiobjec-
tive fuzzy regulator design is shown in Table 1. The
desired closed loop poles are chosen to be allocated
for each submodel in the consequent proposition of
the Takagi-Sugeno fuzzy model, which represents the
nonlinear plant. Next, the characteristic equation ac-
cording to desired poles are determined. The charac-
teristic equation is compared with equation (24) and
solved in terms of the variables µ and β. Once that
positive real solutions of µ and β for all submodels
exist, then the state feedback gain matrix can be de-
termined from the equation (22). Otherwise, it is not
possible to allocate the desired poles due to restriction
from the linear quadratic optimal control estrategy.
2.2.2 Stability Conditions based on Lyapunov
Approach
Since fuzzy systems are essentially nonlinear sys-
Table 1: The methodological procedure for the multiobjec-
tive fuzzy regulator design.
Step Procedure
1 Obtain the matrices A
i
and B
i
from linear submodels
2 Choose the poles s
1
and s
2
from feasible regions in Figure 1
Determine µ and β from solution of
3
s
2
+
v
u
u
t
β
(
a
i
1
)
2
β−2a
i
2
β+2
s
β
(
a
i
2
)
2
β+µ
+1
!
β
s+
s
β
(
a
i
2
)
2
β+µ
β
= (s− s
1
)(s− s
2
)
4 Design the state feedback gain matrix K
j
|
[ j=1,2,...,l]
from (22)
tems, stability analysis methods for fuzzy control sys-
tems, in particular, are based on nonlinear stabil-
ity theory. In the literature, some stability analysis
methods based on Lyapunov approach are proposed
(Lendek et al., 2009; Tanaka et al., 1996; Sheik-
holeslam and Shekaramiz, 2011).
Consider a continuous fuzzy control system (CFS)
described by:
˙
x(t) =
l
∑
i=1
l
∑
j=1
λ
i
(x(t))λ
j
(x(t)) [A
i
− B
i
K
j
]x(t)
=
l
∑
i=1
λ
i
(x(t))G
ii
x(t) + (30)
+2
l
∑
i< j
λ
i
(x(t))λ
j
(x(t))
G
ij
+ G
ji
2
x(t)
where,
G
ij
= A
i
− B
i
K
j
(31)
Stability conditions for this CFS are established
by following theorem (Tanaka et al., 1996; Michels
et al., 2006):
Theorem 1. The equilibrium of a CFS is asymptoti-
cally stable in the large if there exists a common pos-
itive definite matrix P such that
G
T
ii
P+ PG
ii
< 0 (32)
G
ij
+ G
ji
2
T
P+P
G
ij
+ G
ji
2
< 0, i < j, (33)
for all i and j excepting the pairs (i, j) such that
λ
i
(x(t))λ
j
(x(t)) = 0 for all t.
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
54
3 COMPUTATIONAL RESULTS
This section presents the computational results show-
ing the efficience of the proposed methodology from
the multiobjective fuzzy regulator design for a robotic
manipulator.
The differential equation of the robotic manipula-
tor is given by:
ml
2
¨
θ+ Bl
˙
θ+ mgl sin(θ) = T
c
(34)
where B = 1kgm
2
/s is the damping factor, m = 1kg
is the mass and l = 1m is the length of the manipu-
lator arm; g = 9.81m/s
2
is the gravitational constant,
T
c
is the input variable, i.e., the torque in N.m. The
angular position θ is the output variable of the manip-
ulator. Consider an angular position θ
0
of the robotic
manipulator. Equation (34) can be formulated by:
¨
θ = −ψ
˙
θ− δθ+ u
∗
(35)
where α =
1
ml
2
, ψ =
B
ml
, γ(θ
0
) =
g
l
[sin(θ
0
) −
θ
0
cos(θ
0
)], δ(θ
0
) =
gcos(θ
0
)
l
and u
∗
= αT
c
−γ. The
state space representation is given by
˙x(t) =
0 1
−δ(θ
0
) −ψ
x(t) +
0
1
u
∗
y(t) =
1 0
x(t)
(36)
where x
1
(t) = θ, x
2
(t) =
˙
θ, ˙x(t) =
˙x
1
(t)
˙x
2
(t)
, x(t) =
x
1
(t)
x
2
(t)
and y(t) is the angular position of the
robotic manipulator.
Considering operating points for the robotic ma-
nipulator −90
◦
, −60
◦
, −30
◦
, 0
◦
, 30
◦
, 60
◦
, and 90
◦
,
as shown in Figure 2, it is possible to obtain the linear
submodels and group them into a TS fuzzy structure.
The rule base for TS fuzzy model is shown in Table
2.
Figure 2: Operating points of the manipulator.
Table 2: Rule base of the Takagi-Sugeno fuzzy model for
robotic manipulator described by the equation (34).
Model
R
1
: If θ is −90
◦
then ˙x(t)=
"
0 1
0 −1
#
x(t)+
"
0
1
#
u
1
∗
R
2
: If θ is −60
◦
then ˙x(t)=
"
0 1
−4.9050 −1
#
x(t)+
"
0
1
#
u
2
∗
R
3
: If θ is −30
◦
then ˙x(t)=
"
0 1
−8.4957 −1
#
x(t)+
"
0
1
#
u
3
∗
R
4
: If θ is 0
◦
then ˙x(t)=
"
0 1
−9.81 −1
#
x(t)+
"
0
1
#
u
4
∗
R
5
: If θ is 30
◦
then ˙x(t)=
"
0 1
−8.4957 −1
#
x(t)+
"
0
1
#
u
5
∗
R
6
: If θ is 60
◦
then ˙x(t)=
"
0 1
−4.9050 −1
#
x(t)+
"
0
1
#
u
6
∗
R
7
: If θ is 90
◦
then ˙x(t)=
"
0 1
0 −1
#
x(t)+
"
0
1
#
u
7
∗
The rule base for TS fuzzy regulator is shown in
Table 3. The fuzzy control action is given by:
˜u(t) = −
7
∑
i=1
λ
i
K
i
˜x+
7
∑
i=1
λ
i
u
i
(37)
where λ
i
denotes the normalized fulfillment degree.
Table 3: Rule base of the Takagi-Sugeno fuzzy regulator
for robotic manipulator described by the equation (34). For
this rule base, u
1
= 9.81, u
2
= 3.3592, u
3
= 0.4567, u
4
= 0,
u
5
= −0.4557, u
6
= −3.3592 and u
7
= −9.81.
Controller
R
1
: If θ is − 90
◦
then ˜u(t) = −K
1
˜x(t) + u
1
R
2
: If θ is − 60
◦
then ˜u(t) = −K
2
˜x(t) + u
2
R
3
: If θ is − 30
◦
then ˜u(t) = −K
3
˜x(t) + u
3
R
4
: If θ is 0
◦
then ˜u(t) = −K
4
˜x(t) + u
4
R
5
: If θ is 30
◦
then ˜u(t) = −K
5
˜x(t) + u
5
R
6
: If θ is 60
◦
then ˜u(t) = −K
6
˜x(t) + u
6
R
7
: If θ is 90
◦
then ˜u(t) = −K
7
˜x(t) + u
7
AProposalofMultiobjectiveFuzzyRegulatorDesignforStateSpaceNonlinearSystems
55
The simulation diagram of the robotic manipulator
and fuzzy regulator are shown in Figure 3.
x(t)
x1(t)
x2(t)
x2(t)
x1(t)
u(t)
rad − degree
Converter
180/pi
Robotic
Manipulator
Torque
x1
x2
Offsets
In17
In18
In19
In20
In21
In22
In23
Out15
Out16
Out17
Out18
Out19
Out20
Out21
Normalized
Fulfillment Degree
In2
In3
In4
In5
In6
In7
In8
gama 1
gama 2
gama 3
gama 4
gama 5
gama 6
gama 7
Membership
Functions
In1
Out1
Out2
Out3
Out4
Out5
Out6
Out7
LQR /
Pole Placement
In9
In10
In11
In12
In13
In14
In15
In16
Out8
Out9
Out10
Out11
Out12
Out13
Out14
Figure 3: Simulation diagram of the multiobjective fuzzy
regulator design.
The transient response and the control action of
the multiobjective fuzzy regulator system for the
robotic manipulator, considering some complex and
real poles of the feasible and unfeasible region, from
Table 4, are shown in Figure 4.
Figure 4: Multiobjective fuzzy controller performance. The
poles s
1
= −8, s
2
= −10 (solid line) and s
1
= −3+ j2, s
2
=
−3 − j2 (dash-dot line) are allocated in the feasible re-
gion; the poles s
1
= −0.3, s
2
= −0.5 (dotted line) and
s
1
= −0.2+ j3, s
2
= −0.2− j3 (dashed line) are allocated
in the unfeasible region.
The poles placed into feasible region implies to
a better transient response specifications. This is be-
cause that the proposed multiobjective methodology
formulation allows the choosing the poles according
to desired transient response and satifies the optimal-
ity criterion from LQR method as well.
Table 4: Relationship between performance criteria through
the matrices Q and R, defined by equations (14) and (15),
respectively, and the pole placement method.
Poles θ
0
µ β K
−90
◦
08.258 0.0323 [16.000 07]
−60
◦
05.683 0.0245 [11.095 07]
−30
◦
03.830 0.0208 [07.504 07]
[−4,−4] 0
◦
03.156 0.0198 [06.190 07]
30
◦
03.830 0.0208 [07.504 07]
60
◦
05.683 0.0245 [11.095 07]
90
◦
08.258 0.0323 [16.000 07]
−90
◦
18.778 0.1111 [13.000 05]
−60
◦
07.706 0.0532 [08.095 05]
−30
◦
03.725 0.0385 [04.504 05]
[−3± j2] 0
◦
02.542 0.0349 [03.190 05]
30
◦
03.725 0.0385 [04.504 05]
60
◦
07.706 0.0532 [08.095 05]
90
◦
18.778 0.1111 [13.000 05]
−90
◦
39.264 0.0061 [80.000 17]
−60
◦
36.896 0.0058 [75.095 17]
−30
◦
35.156 0.0056 [71.504 17]
[−8,−10] 0
◦
34.518 0.0055 [70.190 17]
30
◦
35.156 0.0056 [71.504 17]
60
◦
36.896 0.0058 [75.095 17]
90
◦
39.264 0.0061 [80.000 17]
Below it is proved that the state feedback gain
matrices K
i
(i = 1,2,··· ,l) obtained by proposed
methodology for fuzzy regulator design satisfies the
conditions of Theorem 1 for a common positive def-
inite matrix P. Therefore, the procedure for this
demonstration is as follow:
1. Determine K
i
from A
i
and B
i
using the proposed
methodology;
2. Find a common P satisfying the conditions of
Theorem 1.
The results of the Step 1 were shown in Table 4.
The results of the Step 2 are shown in Table 5 and
Table 6.
The four matrices P obtained from algebraic Ric-
cati equation solution are shown in Tables 5 and 6.
They guarantee the multiobjectivefuzzy regulator sta-
bility since satisfy the conditions established in The-
orem 1 for all rules simultaneously. As can be seen
in Table 5 and 6, all values of the matrices P ensure
matrices G
T
ii
P+ PG
ii
|
i=1, 2,···,7
and
G
ij
+G
ji
2
T
P+
P
G
ij
+G
ji
2
whose eigenvalues are negative.
ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics
56
Table 5: The matrix G
T
ii
P+ PG
ii
and its eigenvalues for all
values of the matrix P .
P G
T
ii
P+ PG
ii
|
i=1, 2,···, 7
P =
8.8344 0.4908
0.4908 0.1043
−78.5276 −08.3436
−08.3436 −02.7730
Eigenvalues:
−79.4357 and −1.8649
P =
8.3045 0.4346
0.4346 0.0984
−69.5284 −07.3874
−07.3874 −02.6724
Eigenvalues:
−70.3349 and −1.8658
P =
7.9532 0.3973
0.3973 0.0944
−63.5624 −06.7535
−06.7535 −02.6056
Eigenvalues:
−70.3349 and −1.8658
P =
7.8315 0.3844
0.3844 0.0931
−61.4960 −06.5340
−06.5340 −02.5825
Eigenvalues:
−62.2120 and −1.8666
Table 6: The matrix
G
ij
+G
ji
2
T
P + P
G
ij
+G
ji
2
and its
eigenvalues for all values of the matrix P .
P
G
ij
+G
ji
2
T
P+P
G
ij
+G
ji
2
i, j=1,2,···,7(i< j)
P =
8.8344 0.4908
0.4908 0.1043
−78.5276 −08.3436
−08.3436 −02.7730
Eigenvalues:
−79.4357 and −1.8649
P =
8.3045 0.4346
0.4346 0.0984
−74.0280 −07.8610
−07.8699 −02.7227
Eigenvalues:
−74.8853 and −1.8654
P =
7.9532 0.3973
0.3973 0.0944
−71.0450 −07.5411
−07.5559 −02.6893
Eigenvalues:
−71.8687 and −1.8657
P =
7.8315 0.3844
0.3844 0.0931
−70.0118 −07.4304
−07.4472 −02.6778
Eigenvalues:
−70.8238 and −1.8658
4 CONCLUSIONS
A new fuzzy multiobjective control design method-
ology for nonlinear dynamic systems was proposed
in this paper. In this approach, two techniques, which
are pole placement and LQR methods widely used for
linear systems, were combined and extended for non-
linear systems via state space Takagi-Sugeno fuzzy
inference structure. Simulation results shown that the
multiobjective feasible region allows the choosing the
poles according to desired transient response and sat-
ifies the optimality criterion from LQR method as
well. For further works, the following research in-
terest can be considered:
• Industrial plants applications via high perfor-
mance virtual/electronics instrumentation;
• Adaptive control design, once that analytical for-
mulas for multiobjective fuzzy regulator were ob-
tained;
• Multiobjective fuzzy regulator design for multi-
variable and/or time delayed nonlinear plants.
ACKNOWLEDGEMENTS
The authors would like to thank FAPEMA (Research
and Scientific and Technological Development Sup-
port Foundation of the Maranhão) for financial sup-
port.
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