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International Journal of Control Science and Engineering 2012, 2(5): 120-126
DOI: 10.5923/j.control.20120205.04
PID Studies on Position Tracking Control of an
Electro-Hydraulic Actuator
No rle l a Is hak 1,*, Mazidah Tajjudin1, Has himah Ismail2, Mohd He zri Fazalul Rahiman1, Yahaya Md Sam3,
Ramli Adnan1
1Facult y of Electrical En gineer in g, UiTM , 40450, Shah Alam, Selan gor, M alay sia
2Faculty of Engineerin g, UNISEL, 45600 Bestari Jaya, Selangor, Malaysia
3Facult y of Electrical En gineer in g, UTM Sekudai, 81310 Johor, M alays ia
Abs t ra c t Despite the application of advanced control technique to improve the performance of electro-hydraulic position
control, Proportional Integral Derivative (PID) control scheme seems able to produce satisfactory result. PID is preferable in
industrial applications because it is simple and robust. The main problem in its application is to tune the parameters to its
optimu m values. This study will look into an optimization of PID parameters using Nelder-Mead (N-M) compare with
s elf -tuning fuzzy approach for electro-hydraulic position control system. The electro-hydraulic system was represented by an
Auto-regressive with Exogenous Input (ARX) model structure obtained through MATLAB System Identification Toolbox.
Second-order and third-order model of the system had been evaluated. Simulation and real-time studies show that the output
produced the best response in terms of transient speed and Root Mean Square Error (RMSE) performance criteria.
Ke y wo rds Electro-Hydraulic System, PID Control, Nelder-Mead Optimization, Se lf Tu n in g Control, System
Id en tific at io n
1. Introduction
Electro-hydraulic actuators are very important elements
for industrial processes because th ey provide linear
move ment, fast response and accurate positioning of heavy
load. Recently, hydraulic actuator system has gained
popularity in many applications such as in p aper mil ls ,
aircra fts, an d automotive industries where linear move ment,
fast response, and accurate positioning with heavy loads are
required.
Howe ver, the nonlinear nature of such actuators represents a
hard challenge in designing a perfect controller for this
actuator. Difficulties in identifying an accurate model of
inherently nonlinear and time -varying dynamics ma ke
controller design more complicated. Many researchers have
used advanced control strategies to improve the system
performance mainly in tracking control and motion control
ability. Chen et al.[1] and Ghazali et al.[2] had applied
sliding mode control, many others had applied hybrid of
fuzzy and PID and adaptive PID control using fuzzy[3-6].
Th e ir studies show that the PID control laws are sufficient to
control the hydraulic actuator as desired.
Feedback control system design using PID controller has
* Corresponding author:
norlelaishak@salam.uitm.edu.my(Norl ela Ish ak)
Published online at http://journal.sapub.org/control
Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved
been adopted in this study because it is simple and robust
when applied within specified operating range. The equation
for a typical digital PID controller is given in Eq. (1).
(1 )
Where e(k) is the error signal.
To ensure a good performance of the controller, suitable
values for each parameter namely Kp, Ki and Kd must be
tuned optimally. Classical PID tuning approach such as
Zie gle r -Nichols and Cohen-Coon requires information of
ultimate gain and ultimate period of oscillation in order to
calculate the controller parameters. The disadvantage of
experimentally determining the critical parameters is that the
system can lead to a state of instability. Finding a stability
boundary in systems with large t ime constants can be very
time-consuming[4].
In an effort to improve the performance of PID tuning for
processes with changing dynamic properties, this study will
applied automatic tuning based on Nelder-Meadoptimizatio
n and self-tuning fuzzy to tune the PID para meters. The
optimization algorithm will search for optimal values of Kp,
Ki and Kd from a given specified step response
req uire ments and actuator constraints. The detail will be
explained in s ection optimization PID. The tuning will be
done by simulation. The performance controller will be
evaluated using a sinusoidal signal with time-varying
frequency and demonstrated on a hydraulic position control
test bed.
∑
=
−−++= k
0j )]1k(e)k(e[
d
k)j(e
i
k)k(e
p
k)k(u
121 International Journal of Control Science and Engineering 2012, 2(5): 120-126
2. System Identification
This study was implemented on an electro-hydraulic
s ys tem with s in g le -ended cylinder type of actuator and the
pressurized fluid flow is control by a proportional valve. The
bidirectional cylinder has 150 millimeter stroke length; 40
millimete r bore size and 25 millimeter rod s ize. The wire
displacement sensor is mounted at the top of cylinder rod.
The comp lete experimental setup for data collection and
real-time studies is shown in Figure 1. The data collection for
input-output test of the plants was done using MATLAB
Real-time workshop via Advantech PCI-1716 interface card.
The input signal used for model identification was a
mu lt i -frequency sine waves generated using three different
frequencies as represented by Eq. (2).
Vin (k) = 2 cos 0.3 tsk + 2 cos 4 t sk + co s 6 tsk (2)
where ts is the sampling time
A set of data that consists of the input voltage and actuator
displacement as shown in Figure 2a and 2b was observed for
5000 time steps experiment with 40ms of samp ling time
under the off-line model identification.
Fi gu re 1. Experiment al setup of electro-hydraulic system
Fi gu re 2. Input and Output signal for model ident ification
When the system was perturbed by a signal up to third
harmonics, the model that can be obtained is limited to
second and third order only. Higher orders model may
produce unstable output[7].
For linear identification process , discrete time ARX model
structure was selected for this study primarily to represent
the system for PID controller. The ARX model is a simple
model and can be presented in a simple linear d ifference
equation. In this study, second-order and third-order A RX
model had been estimated with best fit of more than 80%.
Four possible models were obtained as tabulated in Table 1.
All models are stable and of minimum phas e as can be
evaluated from the location of its poles and zeros. The
pole-ze ro ma p s are given in Figure 3.
Table 1. ARX model repr esent at ion with best fit crit eri a
Mo de l
Orde r
Po lyno m ial Be st f it
ARX211 A(q) = 1-1. 795 q
-1
+0.7954 q
-2
B(q) = 0.0025 69q
-1
82. 03 %
ARX221 A(q) = 1-1.879q
-1
+0.8796 q
-2
B(q) = 0.01088q
-1
-0.009358q
-2
84. 24 %
ARX311 A(q) = 1-2.173q-1+1. 553 q-2 -0.38q-3
B(q) = 0.0023 79q-1 86. 53 %
ARX331 A(q) = 1-2.099q-1+1. 35 q-2 -0.2505q-3
B(q) =0.00621 3 q-1-0.002792q-2-0.001707q-3 87.57%
Based on best fit performance criteria as expected,
ARX331 is the best model to represent the system. Generally,
model representation with adequate accuracy is required in
order to design a controller that will drive the output in a
desired manner[8]. This study will determine how accurate
the model would be considered as adequate model for PID
control implementation.
Fi gu re 3. Po le – Zero Map
3. PID Optimization
In this study, the PID parameters will be optimized using
Nelder-Mead optimization and self-tuning Fuzzy approach.
The Nelder-Mead technique was proposed by Nelder and
Mead in 1965[9]. It is a s imp le x-based method to find a local
minimu m of a function of several variables . It attempts to
minimize a nonlinear function of n variables without any
derivative information. This method applied a pattern search
approach with k+1dimensional shape where k is the number
Norlela Ishak et al.: PID Studies on Position Tracking Cont rol of an Electro-Hy draulic Actuator 122
of variables to be optimized. Along the search, the initial
simplex (polygon) will go through a process of reflection,
expansion, contraction and shrinking until the function is
minimized (or ma ximized) . The procedure of Nelder-Mead
search is lis t ed in Table 2.
Table 2. N-M algor ithm for 3 p arameters
IF f(R) < f(G), THEN perform Case(i) {either reflect or extend}
ELSE perform Case (ii) {either contract or sh rink}
BEGIN {Case(i)}
IF f(B) < f(R) THEN
Replace W with R
ELSE
Compute E and f(E)
IF f(E) < f(B) THEN
Replace W with E
ELSE
Replace W with R
E NDI F
E NDI F
END {Case(i)}
BEGIN {Case(ii)}
IF f(R) < f(W) THEN
Replace W with R
Co mp ut e C = ( W + M)/2
Or C = (M + R)/2 and f(C)
IF f(C) < f(W) THEN
Replace W with C
ELSE
Compute S and f(S)
Replace W wit h S
Replace G with M
E NDI F
END {Case(ii)}
Nelder-Mead optimization still attracts res earche r fro m
many areas even though it seems too colloquial[10-12]. It is
a close relative to Particle Swarm Optimization (PSO) and
Differential Evolution (DE)[13]. Wang et al.[14] applied
this method for parameter estimat ing of chaotic system and
Panigrahi and Pandi[15] applied Nelder-Mead along with
Bacterial Foraging Optimization (BFO) to explore the
search space to find the local minima for load
dispatch .These shown that it is still the method of choice
for many practitioners in optimization.
In this study, Nelder-Mead is applied to find the optimu m
value for Kp, Ki and Kd with the following constraints:
Rise time: 5s ec
Settling time: 10sec
%Overshoot: 10%
Actuator constraint: ±5V
The controller will be optimized based on step response
specifications within the limited range of controller output
which is ±5V. The system is required to operate at fast
transient with minimu m overshoot. The specifications given
are the best that the optimization could perform.
The PID controller was optimized for all the identified
models. Based on simu lation, the closed -loop output with the
optimized PID controller is shown in Figure 4. From the
figure, there are significant speed variations during transient
response where ARX221 g ive s the h ig hest speed followed
by ARX311. But ARX311 response had slight overshoot
which in some cases may not be to le ra b le becaus e it will lead
to increased s t ead y -state error when apply into the proposed
controller. In this study, the best response in terms of speed
and overshoot was obtained from ARX221 and ARX331
model. The optimized PID shows satisfactory results where
all the outputs lie within the boundaries. The PID parameters
for each model are tabulated in Table 3. Based on the
selected model, parameters of Kp, Ki and Kd will be tested in
simulation and real-time into proposed controller.
Table 3. Optimized Pid Controller Paramet ers (N-M)
ARX211
ARX221
ARX311
ARX331
Kp
2.9030
5
4.3130
4
Ki
0.1 303
0.5
0.0 988
0.2 383
Kd
0.1 954
0
0.0 021
0.0 380
REF
ARX311
ARX211
ARX331
ARX221
Fi gu re 4. C lo se d-loop response with optimized P ID controller
This study also presents a development and
implementation of the proposed self-tuning fu zzy P ID
controller in controlling the position variation of
electro-hydraulic actuator. The self-tuning fuzzy PID
controller is the combination of a classical PID and fuzzy
controller. Self-tuning fuzzy PID controller means that the
three parameters Kp, Ki and Kd of PID controller are tuned by
using fuzzy tuner[16-17]. The coefficients of the
conventional PID controller are not properly tuned for the
nonlinear plant with unpredictable parameters variations.
Hence, based on Nelder-Mead optimization parameters , it is
necessary to automatically tune the PID parameters.
Fi gu re 5. St r uct ur e of self-t uni ng f uzzy P ID cont roll er
In this study, the proposed structure of the self-tuning
fuzzy PID controller is shown in Figure 5. There are two
inputs to the fuzzy logic inference engine, the feedback error
e(t) and the derivative of error de(t)/dt. The PID parameters
are tuned by using fuzzy inference, which provide a
nonlinear mapping from the error and derivative of error to
PID parameters.
The rules designed are based on the characteristics of the
electro-hydraulic actuator and properties of the PID
controller. Therefore, the fuzzy reasoning of fuzzy sets of
outputs is gained by aggregation operation of fuzzy sets
10 15 20 25
0
0.5
1
1.5
2
2.5
3
Time Steps
Displacem ent (inch)
123 International Journal of Control Science and Engineering 2012, 2(5): 120-126
inputs and the designed fuzzy rules. The aggregation and
defuzzyfication method are used respectively max-min and
centroid method. Regarding to fuzzy structure, there are two
inputs to fuzzy inference: error e(t) and derivative of error
de(t), and three outputs for each PID controller parameters
respectively K’p, K’i and K’d. Mamdani model is applied as
structure of fuzzy inference with some modification to
obtain the best value for Kp, Ki and Kd. This is illu s trat e b y
Figure 6.
Fi gu re 6. Fuz zy inf eren ce block
The range of each parameter was determined based on the
Nelder-Mead optimization PID controller testing that had
been conducted earlier. This part is important so that a
feasible rule base with high frequency efficiency is obtained.
The ranges of each parameters are :
Kp Є[1, 10] ; Ki Є[0 , 1] ; Kd Є[ 0, 0. 5 ].
Therefore, they can be calibrated over the interval[0, 1] as
follows:
Hence, we obtain : Kp = 9K’p +1 ; Ki = K’i ; Kd = 0.5K’d
The membership functions of the inputs and outputs are
shown in Figure 7a and 7b. Generally, the fuzzy rules are
dependent on the control purpose and type of input-out put
signal parameter. Based on the membership function in
Figure 7a and 7b, the fuzzy rules system was performed as
given in Table 4. The linguistic variables used were Small(S),
Medium Sma ll(MS ), Mediu m (M), Med iu m Big (MB), and
Big (B). Since there we re five lin gu is tic variables that had
been set, thus , 25 fuzzy rules were applied in the system.
Centroid method defuzzification was used to get the definite
values that were sent to PID controller. The whole systems
were developed using Matlab Simu link environment.
Table 4. Rules of the fuzzy in ference
de/dt
Er ro r (e)
NB
NS
ZE
PS
PB
NB
S
S
MS
MS
M
NS
S
MS
MS
M
MB
ZE
MS
MS
M
MB
MB
PS
MS
M
MB
MB
B
PB
M
MB
MB
B
B
Fi gure 7 a. M em bershi p f unct ion o f e(t ) an d de(t )
These levels are chosen from the characteristics and
specification of the electro-hydraulic actuator. Figure 7a,
shows the ranges of these inputs are -0.1 to 0.1 and -.01 to 0.3,
which are obtained from the absolute value of the system
error and its derivative through the gains.
Figure 7b, shows the ranges of outputs K’p, K’i and K’d
where the ranges from 1 to 10, 0 to 1 and 0 to 0.5.
Figure 7 b. Membership function of K’p, K’i and K ’d
110
1K
KK
KK
K
p
minpmax
p
minpp
'
p
−
−
=
−
−
=
01
0K
KK
KK
K
i
minimaxi
minii
'
i
−
−
=
−
−
=
05.0
0K
KK
KK
K
d
min
dmaxd
mindd
'
d
−
−
=
−
−
=
Norlela Ishak et al.: PID Studies on Position Tracking Cont rol of an Electro-Hy draulic Actuator 124
4 .Simulation and Real-Time
Implementatio n
The N-M optimized PID setting was used to simulate the
system performance when subjected to step response and
reference sinusoidal signal with time-varying frequency.
Root mean square error (RMS E) was selected as the
performance criteria. Tab le 5 s ummarizes the performance
of the identified models. Based on RMSE index, ARX221
and ARX331 model has outperformed other models.
Table 5. RM SE performance criteria fo r s im u l at ion N e l der -Mead
RMSE (i nch)
ARX211
ARX221
ARX311
ARX331
U n it st e p
0.2 935
0.2 568
0.2 880
0.2 761
Re f Si n e
0.1 634
0.1 162
0.1 501
0.1 356
The ARX221 and ARX331 model then evaluated towards
reference sinuso id al s ignal with time-varying frequency. The
shape is chosen such that to demonstrated the ability of the
controller to track the reference signal with changing
frequency components. Figure 8 shows the output response
for t he sinusoidal signal with time-varying frequency for
Nelder-Mead optimization.
REF
ARX221
ARX331
Fi gu re 8. Sim ulat ion resul t of the sinuso ida l re spon se s of N -M PID
Based on the Nelder-Mead optimization, we proposed
s elf -tuning fuzzy PID controller in controlling position
variation of electro-hydraulic actuator. The parameters of
each controller have been optimized based on Nelder-Mead
algorithm. In order to perform the output of the system, two
types of input signal are applied respectively step input and
sinusoidal input with time-varying frequency. For
comparison purposes, the root mean square error (RMSE)
was selected as the performance criteria . Table 6 shows the
overall results during simulation. The outputs of simulation
for Ne lder-Mead optimization and self-tuning fuzzy control
are presented in Figure 9, 10, 11 and 12 below.
Table 6. RM SE performance criteria fo r simulat ion Nelder -Mead
RM SE
( in ch )
ARX221
ARX331
N-M P I D
FUZZY
PID
N-M P I D
FUZZY
PID
U n it st e p
0.2 568
0.2 532
0.2 761
0.2 623
Re f Si n e
0.1 162
0.0 813
0.1 356
0.0 843
Based on the error analyses, control effort and observation
on the tracking performance, the self-tuning fuzzy control
provides more convenient and better performance in position
tracking control. Co mpare with the Nelder-Mead PID
control strategy, the self-tuning fuzzy PID controller reduced
the error. This can observed from the RMSE index given in
Table 6.
Fi gu re 9. Simulation result of the step responses of N-M an d Fuzzy P ID
(ARX221)
Figure 10 . Simulat ion result of the step r espon ses of N-M an d Fuzz y PID
(ARX331)
REF
ARX 221
N-M PID
ARX 221
FUZZY PID
Figure 11 . Simulat ion re sult of t h e sinusoidal r esponses of N -M an d
Fuzz y PI D ( ARX221)
REF
ARX33 1
N-M PID
ARX33 1
FUZZY PID
Figure 12 . Simulat ion re sult of t h e sinusoidal r esponses of N -M an d
Fuzz y PI D ( ARX331)
50 100 150 200 250 300
0.5
1
1.5
2
2.5
3
3.5
4
Time Steps
Displac ement (inc h)
10 15 20 25
0
0.5
1
1.5
2
2.5
3
Time Steps
Displac ement (inc h)
Ref
ARX221 NM PID
ARX221 FUZZY PID
10 15 20 25
0
0.5
1
1.5
2
2.5
3
Time Steps
Displac ement (inc h)
Ref
ARX331 NM PID
ARX331 FUZZY PID
50 100 150 200 250 300
0.5
1
1.5
2
2.5
3
3.5
4
Time Steps
Displac ement (inc h)
50 100 150 200 250 300
0.5
1
1.5
2
2.5
3
3.5
4
Time Steps
Displac ement (inc h)
125 International Journal of Control Science and Engineering 2012, 2(5): 120-126
(a) Po sit io n t racking
(b) Posit ion tracking error
Figure 1 3. Exper iment al result of the step re spon se s of N-M and Fuz zy
PID (ARX221)
(a) Position tracking
(b) Posit ion tracking error
REF
N-M P I D
FUZZY PID
Figure 14 . Exper imental result of the sinusoidal respon ses of N-M a n d
Fuzz y PI D ( ARX221)
Figure 9 and 10 shows the output response of both the
N-M and Fuzzy PID controllers. It can be seen that both
controllers satisfactorily reaches the steady-state without
ov ershoot . A faster ris e-time and settling-time are recorded
in the Fuzzy PID response obtained from ARX221 model
with PID setting Kp= 5, Ki = 0.5 and Kd = 0. F ig u re 11 and
12 shows resulting tracking when using sinusoidal responses.
Figure 11 shows much better response that the one given in
figure 12. In fact, the overlapping of reference and output
signals cannot be seen.
Therefore, the simulation work was verified by applying
the controller parameters (Kp, Ki and Kd) of ARX221 model
to real system to achieve the best performance of the system.
Hence, the results showed that the output of the system with
the design controller by simulation and experiment were
improved and almost similar.
Fig ure 13 shows the experimental output response of N-M
and Fuzzy PID controllers. It can be seen that the Fuzzy PID
satisfactorily reaches the s teady-state without overshoot and
reduced the error compare with N-M PID.
Fro m Figure 14, the experimental result shows that N-M
PID control responses has serious delay and large tracking
error, while the response speed and tracking accuracy of
s elf -tuning fuzzy PID control is better.
5. Conclusions
This study had imp lemented Nelder-Mead optimization to
tune the PID parameters for a given constraints of desired
step response. A self-tuning fuzzy PID controller was
successfully developed and applied to the electro-hydraulic
actuator using the parameters that have been optimized
earlier by Nelder-Mead algorithm. The robustness and
effectiveness of the designed controllers were verified
through computer simulations and experiments. The results
show that self-tuning fuzzy PID controller seems feasible to
control the electro-hydraulic according to desired reference
signal. The proposed controller offers promising capabilities
to guarantee the robustness and position tracking accuracy of
the system. The pos ition tracking performance was imp roved
by using controller parameters value of Kp, Ki and Kd for
second-order model.
ACKNOWLEDGEMENTS
The authors would like to thanks and acknowledge the
FRGS -RMI-U iTM (600-RMI/ST/FRGS/5/3/Fst(85/2010)
for financial support of this research work.
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10 15 20 25
0
0.5
1
1.5
2
2.5
3
Time Steps
Displac ement (inc h)
Ref
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3
Time Steps
Displac ement (inc h)
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