Design Mathematical Tunable Gain PID-Like Sliding Mode Fuzzy Controller With Minimum Rule Base

Farzin Piltan, N. Sulaiman, A. Gavahian, S. Soltani & S. Roosta
International Journal of Robotics and Automation (IJRA), Volume (2) : Issue (3) : 2011 146
Design Mathematical Tunable Gain PID-Like Sliding Mode Fuzzy
Controller with Minimum Rule base
Farzin Piltan SSP.ROBOTIC@yahoo.com
Department of Electrical and Electronic
Engineering, Faculty of Engineering,Universiti
Putra Malaysia 43400 Serdang, Selangor, Malaysia
N. Sulaiman nasri@eng.upm.edu.my
Department of Electrical and Electronic
Engineering, Faculty of Engineering,Universiti
Putra Malaysia 43400 Serdang, Selangor, Malaysia
Atefeh Gavahian SSP.ROBOTIC@yahoo.com
Industrial Electrical and Electronic Engineering
SanatkadeheSabze Pasargad. CO (S.S.P. Co),
NO:16 ,PO.Code 71347-66773, Fourth floor Dena
Apr , Seven Tir Ave , Shiraz , Iran
Samira Soltani SSP.ROBOTIC@yahoo.com
Samaneh Roosta SSP.ROBOTIC@yahoo.com
Abstract
In this study, a mathematical tunable gain model free PID-like sliding mode fuzzy controller (GTSMFC) is
designed to rich the best performance. Sliding mode fuzzy controller is studied because of its model free,
stable and high performance. Today, most of systems (e.g., robot manipulators) are used in unknown and
unstructured environment and caused to provide sophisticated systems, therefore strong mathematical
tools (e.g., nonlinear sliding mode controller) are used in artificial intelligent control methodologies to
design model free nonlinear robust controller with high performance (e.g., minimum error, good trajectory,
disturbance rejection). Non linear classical theories have been applied successfully in many applications,
but they also have some limitation. One of the best nonlinear robust controller which can be used in
uncertainty nonlinear systems, are sliding mode controller but pure sliding mode controller has some
disadvantages therefore this research focuses on applied sliding mode controller in fuzzy logic theory to
solve the limitation in fuzzy logic controller and sliding mode controller. One of the most important
challenging in pure sliding mode controller and sliding mode fuzzy controller is sliding surface slope. This
paper focuses on adjusting the gain updating factor and sliding surface slope in PID like sliding mode
fuzzy controller to have the best performance and reduce the limitation.
Keywords: Sliding Mode Fuzzy Controller, Tunable Gain, Artificial Intelligence, Robust Controller, Sliding
Mode Controller, Fuzzy Logic Theory, Sliding Surface Slope

1. INTRODUCTION
The aim of science and modern technology has making an easier life. Conversely, modern life includes
complicated technical systems which these systems are nonlinear, time variant, and uncertain in
measurement, they need to have controlled [2]. Controller (control system) is devices that can sense data
from plant to improve the plants behavior through actuation and computation. Fuzzy logic theory was used
in wide range applications that fuzzy logic controller (FLC) is one of the most important applications in
fuzzy logic theory. Conversely pure FLC works in many areas, it cannot guarantee the basic requirement
of stability and acceptable performance [4-5].
Sliding mode controller (SMC) is one of the influential nonlinear controllers in certain and uncertain
systems which are used to present a methodical solution for two main important controllers’ challenges,
which named: stability and robustness. Conversely, this controller is used in different applications; sliding
mode controller has subsequent drawbacks i.e. chattering phenomenon, and nonlinear equivalent dynamic
formulation in uncertain systems[6-12].
Although both SMC and FLC have been applied successfully in many applications but they also have
some limitations. The boundary layer method is used to reduce or eliminate the chattering [1, 3, 12] and
proposed method focuses on applied sliding mode controller to proposed PID error-base fuzzy logic
system with minimum rule base and adjust the sliding surface slope to implement easily and avoid
mathematical model base controller.
This paper is organized as follows:
In section 2, Detail of classical sliding mode controller is presented. The main subject of fuzzy logic
methodology is presented in section 3. In section 4, the proposed method is presented. Modelling PUMA-
560 robot manipulator formulation is presented in section 5. In section 6, the simulation result is presented
and finally in section 7, the conclusion is presented.
3. CLASSICAL SLIDING MODE CONTROL
Sliding mode controller (SMC) is a powerful nonlinear controller which has been analyzed by many
researchers especially in recent years. The sliding mode control law divided into two main parts [1, 3];
࣎ො ൌ ࣎ොࢋࢗ ൅ ࣎ොࢊ࢏࢙ (1)
Where, the model-based component ࣎ොࢋࢗ is compensated the nominal dynamics of systems and ࣎ࢊ࢏࢙ is
discontinuous part of sliding mode controller and it is computed as [16-18]
࣎ොࢊ࢏࢙ ൌ ࡷ. ࢙ࢍ࢔ሺࡿሻ (2)
A time-varying sliding surface ࡿ is given by the following equation [18]:
࢙ሺ࢞, ࢚ሻ ൌ ሺ
ࢊ
ࢊ࢚
൅ ࣅሻ ‫܍‬ ൌ ૙
(3)
Where λ is the constant and it is positive. To further penalize tracking error integral part can be used in
sliding surface part as follows:
࢙ሺ࢞, ࢚ሻ ൌ ሺ
ࢊ
ࢊ࢚
൅ ࣅሻ ቆන ‫܍‬
࢚
૙
ࢊ࢚ቇ ൌ ૙
(4)
The main target in this methodology is keep ࢙ሺ࢞, ࢚ሻ near to the zero when tracking is outside of ࢙ሺ࢞, ࢚ሻ. The
function of ࢙ࢍ࢔ሺࡿሻ defined as;

࢙ࢍ࢔ሺ࢙ሻ ൌ ൝
૚ ࢙ ൐ 0
െ૚ ࢙ ൏ 0
૙ ࢙ ൌ ૙
(5)
The ࡷ is the positive constant. One of the most important challenges in sliding mode controller based on
discontinuous part is chattering phenomenon which can caused to oscillation in output. To reduce or
eliminate the chattering it is used the boundary layer method; in boundary layer method the basic idea is
replace the discontinuous method by saturation (linear) method with small neighborhood of the switching
surface. This replace is caused to increase the error performance.
࡮ሺ࢚ሻ ൌ ሼ࢞, |ࡿሺ࢚ሻ| ൑ ‫׎‬ሽ; ‫׎‬ ൐ 0 (6)
Where ‫׎‬ is the boundary layer thickness. Therefore, to have a smote control law, the saturation function
ࡿࢇ࢚ሺࡿ
‫׎‬ൗ ሻ added to the control law: Suppose that ࢙࣎ࢇ࢚ is computed as [16-18]
࣎ො࢙ࢇ࢚ ൌ ࡷ. ࢙ࢇ࢚ ቀࡿ
‫׎‬ൗ ቁ (7)
Where ࡿࢇ࢚ ቀࡿ
‫׎‬ൗ ቁ can be defined as
࢙ࢇ࢚ ቀࡿ
‫׎‬ൗ ቁ ൌ
‫ە‬
ۖ
‫۔‬
ۖ
‫ۓ‬ ૚ ሺ࢙
‫׎‬ൗ ൐ 1ሻ
െ૚ ቀ࢙
‫׎‬ൗ ൏ 1ቁ
࢙
‫׎‬ൗ ሺെ૚ ൏ ࢙
‫׎‬ൗ ൏ 1ሻ
(8)
Moreover by replace the formulation (7) in (1) the control output is written as;
࣎ො ൌ ࣎ොࢋࢗ ൅ ࡷ. ࢙ࢇ࢚ ቀࡿ
‫׎‬ൗ ቁ ൌ ൝
࣎ࢋࢗ ൅ ࡷ. ࢙ࢍ࢔ሺࡿሻ , |ࡿ| ൒ ‫׎‬
࣎ࢋࢗ ൅ ࡷ. ࡿ
‫׎‬ൗ , |ࡿ| ൏ ‫׎‬
(9)
4. PID FUZZY LOGIC CONTROLLER
A PID fuzzy controller is a controller which takes error, integral of error and derivative of error as inputs.
Fuzzy controller with three inputs is difficult to implementation, because it needs large number of rules, in
this state the number of rules increases with an increase the number of inputs or fuzzy membership
functions [4-5, 24-31]. In the PID FLC, if each input has 7 linguistic variables, then 7 ൈ 7 ൈ 7 ൌ 343 rules
will be needed. The proposed PID FLC is constructed as a parallel structure of a P+D sliding surface slope
and P+I+D sliding surface slope, and the output of the PID FLC is formed by adding the output of two fuzzy
control blocks. This work will reduce the number of rules needed to 7 ൈ 7 ൌ 49 rules only.
This controller has two inputs (ܵଵ, ܵଶ) and one output ( ߬௙௨௭௭௬). The inputs are first sliding surface (ܵଵ)
which measures by the equation (3), the second sliding surface (ܵଶ) which measures by the equation (4).
For simplicity in implementation and also to have an acceptable performance the triangular membership
function is used. The linguistic variables for first sliding surface (ܵଵ) are; Negative Big (NB), Negative
Medium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), Positive Big
(PB), and it is quantized in to thirteen levels represented by: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 the
linguistic variables for second sliding surface (ܵଶ) are; Fast Left (FL), Medium Left (ML), Slow Left
(SL),Zero (Z), Slow Right (SR), Medium Right (MR), Fast Right (FR), and it is quantized in to thirteen
levels represented by: -6, -5, -0.4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 and the linguistic variables to find the output
are; Large Left (LL), Medium Left (ML), Small Left (SL), Zero (Z), Small Right (SR), Medium Right (MR),
Large Right (LR) and it is quantized in to thirteen levels represented by: -85, -70.8, -56.7, -42.5, -28.3, -

14.2, 0, 14.2, 28.3, 42.5, 56.7, 70.8, 85. Design the rule base of fuzzy inference system can play important
role to design the best performance of sliding mode fuzzy controller, this paper focuses on heuristic
method which, it is based on the behavior of the control systems.
The complete rule base for this controller is shown in Table 1. Rule evaluation focuses on operation in the
antecedent of the fuzzy rules in sliding mode fuzzy controller. Max-Min aggregation is used in this work
which the calculation is defined as follows;
ࣆࢁሺ࢞࢑, ࢟࢑, ࢁሻ ൌ ࣆ‫ڂ‬ ࡲࡾ࢏࢘
࢏స૚
ሺ࢞࢑, ࢟࢑, ࢁሻ ൌ ࢓ࢇ࢞ ቄ࢓࢏࢔࢏ୀ૚
࢘
ቂࣆࡾ࢖ࢗ
ሺ࢞࢑, ࢟࢑ሻ, ࣆ࢖࢓
ሺࢁሻቃቅ (10)
The last step to design fuzzy inference in sliding mode fuzzy controller is defuzzification. In this design the
Center of gravity method ሺ‫ܩܱܥ‬ሻ is used and calculated by the following equation;
࡯ࡻࡳሺ࢞࢑, ࢟࢑ሻ ൌ
∑ ࢁ࢏ ∑ .ࣆ࢛ሺ࢞࢑,࢟࢑,ࢁ࢏ሻ࢘
࢐స૚࢏
∑ ∑ .ࣆ࢛ሺ࢞࢑,࢟࢑,ࢁ࢏ሻ࢘
࢐స૚࢏
(11)
TABLE 1: Modified Fuzzy rule base table
This table used to describe the dynamics behavior of sliding mode fuzzy controller. Table 2 is shown the
COG deffuzzification lookup table in fuzzy logic controller. These output values were obtained by
mathematical on line tunable gain adjustment to reach the best performance in sliding mode fuzzy
controller.
TABLE 2 : COG lookup table in fuzzy sliding mode controller
ࡿ૛
ࡿ૚
Membership Function
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-6 -85 -84.8 -84.8 -84 -82.1 -81 -79 -71 -68 -65 -62 -60 -54
-5 -84.8 -84 -82 -80 -78 -77 -74 -70 -64 -60 -56 -54 -47
-4 -78 -73 -70 -68 -64 -61 -60 -57 -55 -50 -47 -40 -38
-3 -70 -60 -58 -51 -42 -38 -34 -33 -31 -29 -28.4 -28.1 -28
-2 -50 -48 -45 -40 -38 -34 -32 -30 -28 -26 -25 -21 -20
-1 -30 -25 -21 -18 -16 -14 -10 -9 -8 -7 -6.8 -6 -5
0 -10 -8 -6 -1 2 3 6 7 8 10 12 15 17
1 15 18 21 22 23 25 27 28 29 30 30.5 30.8 31
2 29 29.8 31 33 34 34.6 35 35.2 36 37 38 39 42
3 40 41 42 43 45 45 46 46.3 46.8 47 48 51 52
4 48 49 50 52 53 55 56 57 58 59 60 61 63
5 60 61 62 63 64 66 67 68 68.5 69 70 70.8 71
6 66 68.7 68.9 70 72 74 75 77 78 79 81 83 84

5. THE PROPOSED METHOD
Sliding mode controller has two main parts: equivalent controller, based on dynamics formulation and
sliding surface saturation part based on saturation continuous function to reduce or eliminate the
chattering [19-23]. Reduce or eliminate the chattering regarding to reduce the error is play important role in
this research. Applied sliding mode controller in fuzzy logic method have been proposed by several
researchers [19-23] but in proposed method the new PID method with 49 rules is implemented and adjust
by on line mathematical method. SMFC is fuzzy controller based on sliding mode method to easy
implementation, stability, and robustness. A block diagram for sliding mode fuzzy controller is shown in
Figure 1.
FIGURE 1: Sliding Mode Fuzzy Control (SMFC).
The system performance in this research is sensitive to the sliding surface slope ߣ input and output gain
updating factor ‫ܭ‬ఈ & ‫ܭ‬ఉ for sliding mode fuzzy controller. Sliding surface slope can change the response of
the output if large value of ߣ is chosen the response is very fast but the system is very unstable and
conversely, if small value of ߣ considered the response of system is very slow but the system is very
stable. Determine the optimum value of λ for a system is one of the most important challenging works in
SMFC. For nonlinear, uncertain, and time-variant plants on-line tuning method can be used to self
adjusting all coefficients. To keep the structure of the controller as simple as possible and to avoid heavy
computation, a new supervisor tuner based on updated by a new coefficient factor ݇݊ is presented. In this
method the supervisor part tunes the output scaling factors using gain online updating factors. The inputs
of the supervisor term are error and change of error (݁, ݁ሶሻ and the output of this controller is ܷ, which it can
be used to tune sliding surface slope, λ.
࢑࢔ ൌ ࢋ૛
െ
ሺ࢘࢜ െ ࢘࢜࢓࢏࢔ሻ૞
૚ ൅ |ࢋ|
൅ ࢘࢜࢓࢏࢔
(12)
‫ݎ‬௩ ൌ
൫݀݁ሺ݇ሻ െ ݀݁ሺ݇ െ 1ሻ൯
݀݁ሺ. ሻ
݀݁ሺ. ሻ ൌ ൜
݀݁ሺ݇ሻ; ݂݅ ݀݁ሺ݇ሻ ൒ ݀݁ሺ݇ െ 1ሻ
݀݁ሺ݇ െ 1ሻ ݂݅ ݀݁ሺ݇ሻ ൏ ݀݁ሺ݇ െ 1ሻ
ൠ
(13)
In this way, the performance of the system is improved with respect to the SMFC controller. So the new
coefficient is calculated by;
ࣅ࢔ࢋ࢝ ൌ ࣅ࢕࢒ࢊ ൈ ࡷ࢔ (14)
ࡷࢻ࢔ࢋ࢝ ൌ ࡷࢻ࢕࢒ࢊ ൈ ‫ܭ‬݊ (15)

5. APPLICATION
This method is applied to 3 revolute degrees of freedom (DOF) robot manipulator (e.g., first 3 DOF PUMA
robot manipulator). The equation of an n-DOF robot manipulator governed by the following equation [1, 3]:
ࡹሺࢗሻࢗሷ ൅ ࡺሺࢗ, ࢗሶ ሻ ൌ ࣎ (17)
Where τ is actuation torque, ࡹሺࢗሻ is a symmetric and positive define inertia matrix, ܰሺ‫,ݍ‬ ‫ݍ‬ሶሻ is the vector of
nonlinearity term. This robot manipulator dynamic equation can also be written in a following form:
࣎ ൌ ࡹሺࢗሻࢗሷ ൅ ࡮ሺࢗሻሾࢗሶ ࢗሶ ሿ ൅ ࡯ሺࢗሻሾࢗሶ ሿ૛
൅ ࡳሺࢗሻ (18)
Where the matrix of coriolios torque is ࡮ሺࢗሻ, ࡯ሺࢗሻ is the matrix of centrifugal torques, and ࡳሺࢗሻ is the
vector of gravity force. The dynamic terms in equation (15) are only manipulator position. This is a
decoupled system with simple second order linear differential dynamics. In other words, the component ࢗሷ
influences, with a double integrator relationship, only the joint variable‫ݍ‬௜, independently of the motion of the
other joints. Therefore, the angular acceleration is found as to be[3]:
ࢗሷ ൌ ࡹି૚ሺࢗሻ. ሼ࣎ െ ࡺሺࢗ, ࢗሶ ሻሽ (19)
This technique is very attractive from a control point of view. This paper is focused on the design
mathematical tunable gain model free PID-like sliding mode fuzzy controller for PUMA-560 robot
manipulator based on [13-15].
6. RESULTS
PD sliding mode fuzzy controller (PD-SMFC) and mathematical tuneable gain model free PID-like sliding
mode fuzzy controller (GTSMFC) were tested to compare response trajectory. In this simulation the first,
second, and third joints are moved from home to final position without and with external disturbance.
Trajectory performance, chattering phenomenon and disturbance rejection are compared in these two
controllers. These systems are tested by band limited white noise with a predefined 40% of relative to the
input signal amplitude which the sample time is equal to 0.1. This type of noise is used to external
disturbance in continuous and hybrid systems.
Tracking performances: Figure 2 shows the tracking performance in GTSMFC and SMFC without
disturbance for Step trajectory.

FIGURE 2 : Step GTSMFC and SMFC for First, second and third link trajectory without any disturbance.
By comparing, Figure 2, in GTSMFC and SMFC, both of them have the same overshoot (1%) the
GTSMFC and SMFC’s rise time are 0.483 Sec.
Disturbance Rejection
Figure 3 is indicated the power disturbance rejection in GTSMFC and SMFC. A band limited white noise
with predefined of 40% the power of input signal is applied to these controllers; it found slight oscillations in
SMFC’s trajectory responses.

FIGURE 3 : Step GTSMFC and SMFC for First, second and third link trajectory with external disturbance.
Among above graph, relating to trajectory with external disturbance, SMFC has slightly fluctuations. By
comparing overshoot; GTSMFC's overshoot (1%) is lower than SMFC's (2.2%).
Chattering Phenomenon
As mentioned in previous, chattering play important roles in sliding mode controller which one of the major
objectives in this research is reduce or remove the chattering in system’s output with uncertainty and
external disturbance. Figure 4 has shown the power of boundary layer (saturation) method and online
mathematical gain tuning methodto reduce the chattering in GTSMFC and also SMFC.

FIGURE 4: GTSMFC Vs. SMFC chattering with disturbance
7. CONCLUSION
Refer to the research, adjusting the gain updating factor and sliding surface slope in PID like sliding mode
fuzzy controller design and application to robot manipulator has proposed in order to design high
performance nonlinear controller in the presence of uncertainties. Regarding to the positive points in fuzzy
logic controllers in uncertain systems, sliding mode controller which it has stability and robustness and on
line tunable gain to tune the coefficient in structure and unstructured uncertain system the output
responses have improved. Sliding mode controller by adding to the proposed PID fuzzy logic method with
minimum rule base has covered negative points in pure fuzzy logic method and sliding mode methodology.
Obviously the methodology of online tuning is the main goal in this research which most of researcher
used fuzzy logic or neural network to adjust the parameters but in this method we used mathematical
methodology that it is model free.
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Design Mathematical Tunable Gain PID-Like Sliding Mode Fuzzy Controller With Minimum Rule Base

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