10th Portuguese Conference on Automatic Control
16-18 July 2012
CONTROLO’2012
Funchal, Portugal
Gains Tuning of Fuzzy Controllers Based
on a Cost Function Optimization
C. Lucena
∗
∗
L. Palma
A. Cardoso ∗∗ P. Gil
∗,∗∗
∗
Departamento de Engenharia Electrotécnica, Faculdade de Ciências e
Tecnologia, Universidade Nova de Lisboa (e-mail: c.lucena@fct.unl.pt,
psg@fct.unl.pt, lbp@fct.unl.pt).
∗∗
CISUC, Department of Informatics Engineering, University of
Coimbra, Portugal (pgil@dei.uc.pt, alberto@dei.uc.pt)
Abstract: This paper presents a methodology for tuning the gains of fuzzy proportionalintegral controllers where the concept of closed loop control system performance is explicitly
taken into account. The fuzzy controller gains are found by solving a nonlinear constrained
optimization problem considering the system’s dynamics described by a nonlinear model and
a set of constraints on the controller gains, control actions and outputs. Experimental results
from a testbed show the pertinence of using the proposed tuning technique.
Copyright Controlo 2012.
Keywords: Fuzzy Control, Gains Tuning, Nonlinear Optimization, Constrained admissible
Solution.
1. INTRODUCTION
Standard Proportional-Integral-Derivative (PID) control
techniques are still largely used in the process industry,
mainly due to its recognized simplicity. However, in the
case of nonlinear systems it is difficult to get satisfied
closed loop control performance over extended operating
regimes by using classical PID controllers.
Fuzzy Logic Control (FLC) techniques are inherently nonlinear approaches since they incorporate three sources of
nonlinearity, namely, the rule base, the inference engine
and fuzzification and defuzzification modules.
This control paradigm has proven to be successful in
controlling many nonlinear systems and it has been suggested as an alternative approach to conventional control
techniques Feng (2006). FLCs are more robust and their
performance are less sensitive to parametric variations of
systems than conventional controllers Wang et al. (1996).
Furthermore, recent applications of FLC methodologies
have shown great potential in the context of complex illdefined systems, which can effectively be controlled by a
skilled operator without the knowledge of the underlying
systems dynamics Lee (1990).
Nevertheless the great potential of fuzzy control systems
applications, finding a good setting, by choosing appropriate linguistic variables, membership functions, rules and
scaling factors, and subsequently tune it, is still a challenge
issue due to the lack of a systematic framework. Control of
MIMO (Multi-Input/Multi-Output) systems is even more
challenging because one or more manipulated variables
might be affecting the other controlled variables due coupling effects. Ultimately, this interaction might affect the
closed loop system performance.
105
In literature there are multiple approaches to select FLC
parameters. They include heuristics based approaches (see
e.g. Misir et al. (1996)), methodologies relying on pseudoequivalence between digital fuzzy PID controllers and linear PID controllers Boubertakh et al. (2010); Pivonka
(2002) or those based on evolutionary computation Herrera and Lozano (2009).
The present paper proposes a new conceptual methodology
for tuning PI-fuzzy scaling factors or gains, although
not exclusively restricted to this topology, by solving
a nonlinear optimization problem subject to a set of
explicit constraints on scaling factors, control actions and
outputs, and assuming the systems dynamics described by
a nonlinear model. Because the gains are found by solving
an offline constrained optimization problem the closed loop
system response might be considered, to some extent, for
the reference signal adopted in the optimization problem,
as an optimal trajectory provided the certainty equivalence
principle holds.
In order to assess the effectiveness of the proposed approach, it will be compared with a heuristics based controller and a pseudo-equivalence based controller.
2. FUZZY LOGIC CONTROL SYSTEMS
Fuzzy Logic can be defined as a theory of vagueness and
uncertainties. This theory provides an approximate yet
effective mean of describing the behavior of systems, which
could be too complex and ill-defined to permit a precise
mathematical analysis.
The basic structure of FLC (Fig. 1) consists of four conceptual components: knowledge base, fuzzification module,
inference engine and defuzzification module.
Fig. 1. General structure of a Mamdani-type Fuzzy Logic System.
The knowledge base contains all the knowledge of the
fuzzy control system. It comprises a fuzzy control rule
base (procedural part of the knowledge) and a data base
(facts, terms, concepts). The inference engine, regarded as
the kernel of all fuzzy systems, is a reasoning mechanism
that performs inference procedures upon the fuzzy control
rules and given conditions to derive appropriate control
actions. The fuzzification module defines a mapping from a
real-value space to a fuzzy space, while the defuzzification
module implements a mapping from a fuzzy space defined
over an output universe of discourse to a real-valued space
Feng (2006).
One of the major issues regarding fuzzy control design
refers to how to reflect in the tuning stage the concept
of closed loop performance. This would lead to better
fuzzy control systems behavior over extended operation
regimes, while enabling a balance between the tracking
error, energy consumption and actuators wearing. In this
context, the present work proposes an effective structured
automatic framework to deal with the problem of PIfuzzy controller gains tuning based on the minimization
of a user defined explicit metric. This approach not only
makes easier the tuning stage, avoiding cumbersome trialand-error heuristics, but also promotes a more efficient
and rational usage of resources, including energy, which
ultimately contributes to sounding improvements in terms
of overall efficiency and final product quality.
2.1 PI-Fuzzy Controller Structure
With no loss of generality let us consider a two-dimension
PI-fuzzy controller structure, in which error e (1) and
change in error ∆e (2) are selected as input words, while
the output from the fuzzy system is the increment of
control action.
e(k) = r(k) − y(k)
∆e = e(k) − e(k − 1)
(1)
(2)
In the previous equations k is the current discrete time, y
is the system output and r the reference signal.
Fig. 2 shows the PI-Fuzzy control structure where the
scaling factors Ke , K∆e and K∆u , are explicitly represented. The normalized error ẽ, ∆ẽ and the denormalized
increment of control action ∆u are given as:
ẽ(k) = Ke × e(k)
(3)
106
Fig. 2. PI-Fuzzy Logic Control system.
∆ẽ = K∆e × ∆e(k)
(4)
∆u = K∆u × ∆ũ
(5)
After choosing a given setting for the FLC structure the
subsequent step involves the controller tuning. The next
section focuses on the proposed methodology for tuning
the FLC scaling factors based on constrained nonlinear
optimization.
2.2 Optimal Tuning of Scaling Factors
Optimization methods are classified according to the type
of the search space A ⊆ Rn and objective functions. The
simplest method is the linear programming, which concerns
the case where the objective function L(·) to be minimized
is linear and the restriction space A is specified using
only linear equality and inequality constraints Schrijver
(1986). However, in general, the cost function and/or the
constraints contain nonlinearities, leading to a nonlinear
programming problem where multiple minima may exist
in virtue of the nonconvexity of the optimization problem.
This work considers the case in which the performance
index is described by a single objective function subject to
a set of constraints. The underlying optimization problem
can be formulated as follows:
minV ≡ min
K
subject to:
K
Np
X
k=1
Lk (e(k), u(k − 1), K)
(6)
y(k) − g(k) = 0
φ(y(k), u(k), K) = 0
(7)
ψ(y(k), u(k), K) ≤ 0
where V ∈ R+ is the objective function, Np ∈ N + the
prediction horizon, y ∈ Rp the output vector, u ∈ Rm the
control trajectory, K ∈ Rq the vector of scaling factors,
g(·) the nonlinear system dynamics model, φ(·) and ψ(·)
functions defining equality and inequality constraints.
Concerning the underlying system dynamics it is considered a nonlinear model. The main reason is due to the fact
that this kind of models allows a properly approximation
of the system dynamics in a broad range of operating
regimes.
In what the optimization of PI-Fuzzy controller gains
is concerned it is here carried out using the MATLAB
function fmincon() available in the Optimization Toolbox.
The algorithm relies on the Hessian of the Lagrangian (8)
and uses a merit function in the search for the optimal
gains. In each iteration the Hessian matrix is calculated
based on a quasi-Newton approximation.
X
X
52xx L(x, λ) = 52 f (x) +
λi 52 ci (x) +
λi 52 ceqi (x)
(8)
with f the objective function, c the nonlinear inequality
constraint vector, ceq the nonlinear equality constraint
vector and λi the Lagrange multipliers.
3. EXPERIMENTS ON A TESTBED
The testbed used in the experiments consisted of a
R
Feedback
Process Trainer PCT 37-100 (Figure 3). This
system comprises a variable-speed axial fan, regulated
via a potentiometer that circulates an airstream along a
polypropylene tube. The airflow rate is heated by means of
a heating element controlled by a thyristor circuit. A thermistor detector is incorporated to sensing the temperature
at the insertion point.
Table 1. Format of the rule base.
e ∆e
NB
NM
NS
ZE
PS
PM
PB
NB
NB
NB
NB
NB
NM
NP
ZE
NM
NB
NB
NB
NM
NS
ZE
PS
NS
NB
NB
NM
NS
ZE
PS
PM
ZE
NB
NM
NS
ZE
PS
PM
PB
PS
NM
NS
ZE
PS
PM
PB
PB
PM
NS
ZE
PS
PM
PB
PB
PB
PB
ZE
PS
PM
PB
PB
PB
PB
Medium; N S, Negative Small; ZE, Zero; P S, Positive
Small; P M , Positive Medium; P B, Positive Big). For the
fuzzy controller output, ∆ũ, the corresponding universe
of discourse was defined in the range [−1.0 , 1.0], and
assuming a partition similar to those assumed for ẽ and
∆ẽ. The fuzzy inference considered in this work is the
Mamdani-type inference (9), while the controller output is
generated by the Centroid Defuzzification technique. The
membership functions for ẽ and ∆ẽ are presented in Fig.
4 and the membership functions for ∆ũ are shown in Fig.
5
µe/∆e (e, ∆e) = min(µA (e), µB (∆e))
with µ the membership value.
(9)
Concerning the rule base it comprises forty-nine rules, as
shown in Table 1. The design of the rule base was made
according to Li and Gatland (1996).
Fig. 4. Membership functions of ẽ and ∆ẽ.
Fig. 5. Membership functions of ∆ũ.
R
Fig. 3. Feedback
Process Trainer PCT 37-100.
3.2 Experimental Results
3.1 PI-Fuzzy Controller Design
The PI-fuzzy controller comprises two inputs, namely, the
control error and the change in error, and one output,
under the form of control action increment (see Fig. 2).
Regarding the normalized universe of discourse for ẽ
and ∆ẽ they were chosen as [−1.5 , 1.5] and partitioned into seven fuzzy sets, namely, {N B, N M, N S,
ZE, P S, P M, P B} (N B, Negative Big; N M , Negative
107
The experiments included in this section aim to compare the optimal gains PI-fuzzy controller performance
against a gains tuning approach based on a trial-and-error
heuristic and on the pseudo-equivalence between digital
fuzzy PID controllers and linear PID controllers. In these
experiments the airstream temperature should follow a
given user defined reference temperature by manipulating
the input voltage to the heater grid. The default fan speed
was adjusted to position 5 and the sampling time set to
0.1 second.
The first experiment was carried out using a PI-fuzzy
controller with the corresponding gains tuned by means
of a trial-and-error heuristic. As can be observed from
Fig. 6 the closed loop system response is fairly acceptable,
showing a short settling time and no overshoot. However,
this closed loop response was achieved at an expense of a
time consuming and rather tedious tuning process.
The output of the PI-FLC is given by:
∆u(k) = K∆uP I D{F{K∆eP I ∆e(k)+KeP I e(k)}}+u(k−1)
(13)
As can be inferred from this figure, the closed loop system
response is rather oscillatory as a result of an aggressive
controller, whose behavior is reflected in the control signal
variance.
40
45
35
Temperature [Cº]
Temperature [Cº]
45
with T the sampling time, Ti the PI controller time
constant, K the PI controller gain and M the universe
range.
Output
Reference
30
0
10
20
30
40
50
60
70
40
35
Time [s]
Output
Reference
30
5
0
10
20
30
40
50
60
70
50
60
70
Time [s]
4.5
4
Input [V]
3.5
5
3
4.5
2.5
4
2
3.5
Input [V]
1.5
1
0.5
0
0
10
20
30
40
50
60
70
3
2.5
2
1.5
Time [s]
1
Fig. 6. PI-FLC with trial-and-error tuning heuristic.
0.5
Fig. 7 shows the closed loop response and input signals for the pseudo-equivalence between digital fuzzy PID
controllers and linear PID controllers approachPivonka
(2002). According to this approach, the scaling factors of
a PI-Fuzzy controller are given by:
K∆uP I = KT
M
;
Ti
1
KeP I =
;
M
Ti
K∆eP I =
;
M
0
0
10
20
30
40
Time [s]
Fig. 7. PI-FLC with a pseudo-equivalence between digital fuzzy PID controllers and linear PID controllers
tuning approach.
(10)
(11)
(12)
108
The last experiment concerns a context where the PIFuzzy gains are found by solving the following nonlinear
constrained optimization problem, where the cost function
was chosen as a quadratic performance index involving
the penalized control error and the control action, and
assuming the system dynamics described by a nonlinear
model.
K
k=0
2
2
[ y(k) − r(k)] + [0.8u(k − 1)]
)
subject to the system dynamics and
0 ≤ y(k) ≤ 60, k = 1, ..., NP
(14)
(15)
0 ≤ u(k) ≤ 5, k = 0, ..., NP − 1
(16)
0 ≤ Ke
(17)
0 ≤ K∆e
(18)
K∆u ≥ 0
(19)
Temperature [Cº]
min
(N
P
X
45
40
35
Output
Reference
30
10
20
30
40
50
60
70
50
60
70
Time [s]
5
4.5
4
3.5
Input [V]
with u(k) given by:
u(k) = u(k − 1) + K∆u × ∆u(k)
(20)
The nonlinear model of the system is described by a
three-layered feedforward neural network with sigmoidal
activation functions in the hidden layer, and linear activation functions for the output layer. In order to capture
the underlying system’s dynamics the neural network was
trained offline using the Levenberg-Marquardt algorithm,
available in the MATLAB Neural Network Toolbox, and
subsequently validated on a different data set. Equation
(21) describes the system dynamics embedded in the training data set.
0
3
2.5
2
1.5
1



T
6.25 −7.74 −1.16
−1.81
 15.45 −13.38 −2.54
−0.05




y(k) =  25.39  × tangh  7.50 15.35 −5.26
−11.50 2.20 6.53 
 25.15 
−5.92
−0.30 0.13 −0.01


−2.28
"
#
 1.67 
y(k − 1)


× y(k − 2) +  13.74 

−14.52
u(k − 1)
0.35
(21)
Fig. 8 shows the closed loop system response for the offline
gains optimization. As can be observed, the closed system
response is quite satisfactory, exhibiting short settling time
and lacking of any overshoot.
To enable a comparative assessment in terms of performance for the PI-Fuzzy controllers considered in this work
the root mean squared of error (RMSE) given according to
(22) and the root mean squared of control action increment
(RMSI) given according to (23) for each controller are
presented in Table 2 .
v
u PN
T
u
k=1 [y(k) − r(k)] [y(k) − r(k)]
t
RM SE =
PN
T
k=1 r(k) r(k)
(22)
109
0.5
0
0
10
20
30
40
Time [s]
Fig. 8. PI-FLC with offline gains optimization.
RM SI =
q
T
[u(k) − u(k − 1)] [u(k) − u(k − 1)]
N
(23)
As can be inferred from Table 2 the performance of all the
PI-Fuzzy controllers are of the same order of magnitude,
with a slight outperformance for the optimal gains PIfuzzy controller. Although the corresponding closed loop
performances are somehow similar, the ability to automate the tuning process taking into account the closed
performance is invaluable, easing the underlying process
of tuning.
Table 2. Performance metrics.
Tuning Approach
Trial-and-Error
Conventional based PI-Fuzzy
controller
Offline Optimization
RMSE
0.0902
0.0937
Variance
0.0047
0.0095
0.0861
0.0044
4. CONCLUSIONS
This paper addressed the problem of tuning the gains of
Proportional plus Integral fuzzy controllers. The proposed
approach takes into account the closed loop control system
performance and is formulated in terms of a constrained
nonlinear optimization problem. Results collected in a
testbed demonstrated the effectiveness and pertinence of
the proposed methodology, not only in terms of easing and
automating the tuning task, but also in what the closed
loop system performance and longevity of the system is
concerned.
REFERENCES
Hamid Boubertakh, Mohamed Tadjine, Pierre-Yves Glorennec, and Salim Labiod. Tuning fuzzy pd and pi
controllers using reinforcement learning. ISA Transactions, 49(4):543–551, 2010. URL http://dx.doi.org/
10.1016/j.isatra.2010.05.005.
Gang Feng. A survey on analysis and design of modelbased fuzzy control systems. 14(5):676–697, 2006. doi:
10.1109/TFUZZ.2006.883415.
F. Herrera and M. Lozano. Fuzzy Evolutionary Algorithms
and Genetic Fuzzy Systems: A Positive Collaboration between Evolutionary Algorithms and Fuzzy Systems, volume 1 of Intelligent Systems Reference Library. Springer
Berlin Heidelberg, 2009. ISBN 978-3-642-01799-5. URL
http://dx.doi.org/10.1007/978-3-642-01799-5_4.
C. C. Lee. Fuzzy logic in control systems: fuzzy logic
controller. i. 20:404–418, 1990. doi: 10.1109/21.52551.
Han-Xiong Li and H. B. Gatland. Conventional fuzzy
control and its enhancement. 26(5):791–797, 1996. doi:
10.1109/3477.537321.
Dave Misir, Heidar A. Malki, and Guanrong Chen. A
heuristic approach to determine the gains of a fuzzy pid
controller. In Proceedings of the 1996 ACM symposium
on Applied Computing, SAC ’96, pages 609–613, New
York, NY, USA, 1996. ACM. ISBN 0-89791-820-7.
doi: http://doi.acm.org/10.1145/331119.331468. URL
http://doi.acm.org/10.1145/331119.331468.
P. Pivonka. Comparative analysis of fuzzy pi/pd/pid
controller based on classical pid controller approach.
In Fuzzy Systems, 2002. FUZZ-IEEE’02. Proceedings of
the 2002 IEEE International Conference on, volume 1,
pages 541 –546, 2002. doi: 10.1109/FUZZ.2002.1005048.
Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, Inc., New York, NY, USA,
1986. ISBN 0-471-90854-1.
H. O. Wang, K. Tanaka, and M. F. Griffin. An approach to
fuzzy control of nonlinear systems: stability and design
issues. 4(1):14–23, 1996. doi: 10.1109/91.481841.
110