http://dx.doi.org/10.5755/j01.eie.24.2.20630
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 24, NO. 2, 2018
An Adaptive-Fuzzy Fractional-Order Sliding Mode Controller Design for an Unmanned Vehicle Kamil Orman1, Kaan Can2, Abdullah Basci2, Adnan Derdiyok3 Department of Electronics and Automation, Erzincan University, Vocational High School, Erzincan, Turkey 2 Faculty of Engineering, Ataturk University, Electrical & Electronics Engineering, Erzurum, Turkey 3 Faculty of Technology, Sakarya University, Electrical & Electronics Engineering, Sakarya, Turkey
[email protected]
1
with variations and uncertainties depending on parameters under different operating conditions. The proposed sliding surface is defined in fractional proportional-derivative form. Saghafinia et al. [7] have proposed an adaptive fuzzy sliding mode control approach based on the boundary layer approach for speed control with indirect field-oriented control of the induction motor (IM) driver. Experimental and simulation results show the effectiveness of the proposed control structure. Li et al. [8] have designed a new adaptive sliding mode controller for Takagi-Sugeno (T-S) fuzzy systems. The proposed controller that is based on the system output ensures that the closed loop system is properly restrained. Begnini et al. [9], have presented a study on trajectory tracking control of a 2 WD MR. They have used adaptive fuzzy variable structure and PD controllers to realize the kinematic and dynamic controls of the system, respectively. With the proposed controller, the chattering effect of the general variable control approach has been decreased to desired level. Also, the stability of the closed loop control system has been proven via Lyapunov Stability theorem. The simulation and experimental results indicate that the proposed controller has satisfactory performance in terms of reference tracking whereas both controllers have similar error magnitude due to some uncertainty and unknown border layer. Salgado et al. [10], have designed an output feedback controller based on Super Twisting Algorithm (STA) to reduce the trajectory tracking error level of a 4 WD MR. In the structure of the designed control algorithm, the present measurements are the position and drift of the vehicle. By means of the STA that is working depending on the step by step differential system, the speed and acceleration of the vehicle has been estimated. Then, with the second STA, the trajectory tracking control of the vehicle has been realized. Also, the proposed controller is compared with a state output feedback controller and a first order SMC. The experimental results have shown the effectiveness of the proposed controller. In this paper, an AFFOSMC structure has been developed for the control of the speed and direction angle of the mobile
1Abstract—In this paper, speed and direction angle of a four wheel skid-steered mobile robot (4 WD SSMR) has been controlled via adaptive-fuzzy fractional-order sliding mode controller (AFFOSMC) under different speed and angle reference signals. The hybrid control method is designed to combine all advantages that controllers have such as flexibility realized by fractional order calculation, robustness to disturbances and parameters variations provided by sliding mode controller (SMC) as well as adaptation of G constant of SMC via fuzzy controller, simultaneously. Also, a fractionalorder SMC is applied to the system for the same reference speed and angle references to show the effects of the changes and adaptation of G constant. Experimental results show that the AFFOSMC has better trajectory tracking performance than the FOSMC.
Index Terms—Adaptive control; Fuzzy control; Sliding mode control; Fractional-order calculation; UGV.
I. INTRODUCTION Skid steered mobile robots are widely used in industry, transportation, space exploration, military, etc. The trajectory control of skid steered mobile robots has been a problem due to constraints arising from their nonholonomic structures and kinematics. For this reason, in recent years researchers have focused on the motion control of mobile robots [1]–[5]. The sliding mode control (SMC) approach has been successfully applied to mobile robots. The sliding mode control is characterized by simplicity, robustness, reduced system dynamics, limited time convergence, and inherent stability characteristics. However, the biggest problem in the sliding mode control approach is the chattering effect. Chattering effect can induce unmodeled dynamics and cause high frequency oscillations that can even mechanically damage the system. Different approaches have been proposed to reduce this effect. Tang et al. [6] have designed a fractional sliding mode controller that adjusts the desired value of the slip for the ABS braking system, which is used in vehicles and has non-linear system Manuscript received 14 November, 2017; accepted 20 February, 2018.
12
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 24, NO. 2, 2018 robot. The fractional order sliding mode control is also applied to mobile robot in order to compare the performance of the proposed controller. The experimental results show the effectiveness and robustness of the proposed controller.
III. CONTROLLER DESIGN The general control algorithm for this study has been depicted in Fig. 2. The needed torque value for each motor is generated via each control signals uv and u . The torque expression of the right side of the vehicle can be written as
II. DYNAMIC MODEL OF THE 4 WD SSMR In Fig. 1, the two dimensional representation of the system is given [4].
r
uv u . 2
(10)
Also, for the left side is
l
(11)
uv u . 2
The vehicle’s body speed and direction angle is calculated by (2) and (3). Moreover, from (10)–(11), uv and u are generated via the proposed control algorithm for the right and left motors. Fig. 1. Free body diagram of 4 WD SSMR.
The kinematic equations of the system have been derived as given below [11]:
r x 2 v 0 r L
r 2 r 0 , l r L
l vr r , 2 r l , L
r
(1)
Fig. 2. Block diagram of the proposed 4 WD SSMR control system.
(2)
limits of operations. The fractional-order differentiator and integral are defined as follows [12]
A. Fractional-Order Control The fractional-order differentiator can be denoted by a general fundamental operator, a Dtp where a and t are the
(3)
r r , l l ,
(4)
l vx v cos( ) r r cos( ), 2
(5)
l v y v sin( ) r r sin( ). 2
(6)
dp , dt p p a Dt 1, t (dt ) p , a
c
c
l
p 0,
(12)
p 0,
where p is the fractional order which can be a complex
number, however the constant p is related to initial conditions. This operator applied to a f (t ) function leads to the extended Caputo form which is defined as [13]
The restrictions that belong to the vehicle are given as in (7)–(9):
yc cos( ) xc sin( ) a, xc cos( ) y c sin( ) L rr , x cos( ) y sin( ) L r ,
p 0,
t 1 f ( n) ( ) d , (n p ) 0 (t ) p n 1 p a Dt f (t ) dn f (t ), dt n
(7) (8) (9)
n 1 p n, (13)
p n,
where ‘r’ is the diameter of the wheel, ‘2L’ is the distance between right and left wheels, r and l are the angular
where n is the first integer value greater than p and the
velocities of the both wheels, ‘ ’ and ‘ ’ are the vehicle’s body speed and direction angle, respectively, vx and v y
gamma function is defined as ( x ) t x1 e t dt .
0
represent the speed that along the x and y axes, respectively.
Null initial conditions and are taken into account and
13
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 24, NO. 2, 2018 using Laplace transform on D
Laplace a Dtp f (t ) s p F ( s ).
d d ( x) d ( x) dx . dt dt dx dt
(14)
Rewriting (23) and (24) into (20), (25) can be obtained
Fractional order differential equations do not have exact analytical solution; therefore numerical approximation methods such as Oustaloup approximation can be used
s 1 N zn sp k , n 1 1 s
d d (t ) G ( f ( x, t ) Bu ). dt dt
(15)
d dt
B. Sliding Mode Controller Design Before designing the control algorithm, the second order system that the SMC is applied and its parameter error can be defined as given below:
mx(t ) bx (t ) kx(t ) u ,
(16)
(t ) xref (t ) x(t ).
(17)
1 V T 0. 2
(18)
V T .
(19)
(29)
The solution ( x, t ) 0 will be stable if time derivative of
V T D 0,
(20)
(30)
where D is a positive definite matrix. Thus, the time derivative of the Lyapunov function will be negative and this ensures the stability of the system [15]. Also, (29) and (30) lead to
T T D .
(21)
(31)
A solution for (31) is
D 0. (22)
(32)
The expression for derivative of the sliding function is given below
where:
(t ) Gxref ( x) Gx, d ( x) G, dx
(28)
the Lyapunov function can be expressed as
Here, the sliding function can be achieved as in (22)
G ( xref x) G (t ) ( x),
(27)
Taking time derivative of the Lyapunov function, (29) can be written as
1 0 0 x where f ( x , t ) k b and B 1 . Also, the v m m m sliding surface can be defined as given below s x : ( x, t ) 0 .
d (t ) G ( f ( x, t )). dt
(26)
defined positive definite Lyapunov function can be selected as [14]
Using (19), the system can be rewritten as a regular form
f ( x, t ) Bu ,
u ueq
d (t ) G ( f ( x, t ) Bueq ) 0, dt
The equivalent control is valid only if the system is on the sliding surface. Therefore, the system must be driven to sliding surface with another control signal named as reaching signal ur . To obtain this control signal, a well-
Then, with the x v transformation, the system can be rewrite as a first-order system as given below:
x
GBueq
where p 0, N is the number of poles/zeros.
1 (u bv kx ), m 1 0 0 x x k b 1 u . v v m m m
(25)
Then 0 , the equivalent control u ueq is found:
pn
x v
(24)
d (t ) G ( f ( x, t ) Bu ) D 0. dt
,
(23)
(33)
Using (27), (33) can be rewritten as
GBueq GBu D 0.
where G is the design parameter of the system and chosen properly. Taking time derivative of the sliding function, the (24) is achieved
(34)
From (34) and the result of the short algebra, the control signal can be obtained as in (35)
14
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 24, NO. 2, 2018
u ueq K ,
value to realize a robust control. After that, the error and its change with respect to time are both used as an input for a fuzzy set to determine their membership order [17]. According to this membership order of the and , Gˆ (k 1) will be calculated. Moreover, updating Gˆ (k 1)
(35)
where K (GB ) 1 D . During the calculation of ueq given in (28), it is possible to use the equivalent controller estimator principle, which can give an equivalent control effect instead of ueq if B and f(x,t) are not known or if little
f
known. If the physical effect of equivalent control is taken into account, it can be considered that the equivalent control is the average value of the total control. In this case, it is appropriate to design the equivalent control with a low-pass filter which determines the average of the entire signal instead of the rapidly varying high-frequency components in the total control signal [16]. With the proposed method, the control signal can be rewritten as given below
u ueq u K ,
(36)
where u represents the sum of the high frequency part of the u . Therefore, a filter can be designed for the estimated uˆeq as given below
uˆ uˆeq uˆeq ,
f
via an adaptation rule, the ultimate control algorithm has been achieved named as AFFOSMC. The parameters of the fuzzy rules are adjusted by Recursive Least-Square (RLS) algorithm method simultaneously [18]. Also, the control block diagram of the AFFOSMC is given in Fig. 3.
Fig. 3. Control structure of the vehicle.
A linear dynamic system is the output function where i th rule of identifier model is
(37)
~ If x(k ) A i , then Gˆ f i ai x(k ) bi G f (k ),
where
uˆeq
1 u. 1 s
where G f (k ) is the fuzzy design parameter and x(k) is the
(38)
system output. A i is the linguistic value; ai and bi (i = 1, 2, …, R) are the parameter of the results. Gˆ (k 1) is the
In the lighting of this definitions, using (38), (35) can be rewritten as given below
u
1 u K . 1 s
f
identifier model output depending only on the i. rule. Using Centre-Average Defuzzification method, (45) can be achieved
(39)
R
To obtain more robust control strategy, the error and its changes with respect to unit time is taken into consideration. In this case, can be rewritten as in (40)
G .
Gˆ f ( k 1)
1 u K ( G ). 1 s
1 uv K ( a Dtp v G v ), 1 s 1 u u K ( a Dtp G ). 1 s
i 1
R
.
(45)
i
i 1
If the following definition is given for i i
(41)
i
R
.
(46)
i
i 1
The general model can be written as
C. Fractional-Order Sliding Mode Controller Design Applying the same algebra to achieve the FOSMC controller, the control signal equations can be obtained as given below:
uv
Gˆ fi (k 1) i
(40)
Finally, using (40) and substituting this equation in (39) the control signal is found as
u
(44)
Gˆ f ( k 1) T ,
(47)
where Gˆ f (k 1) is the identifier model output, and λ are described as given below
(42)
T
ai ...aR bi ...bR ,
(43)
(48) T
x(k ) i ... x(k )R G f (k )i ... G f (k )R ,
D. AFFOSMC Controller Design In (39), the design parameter G must be has an optimum
(49)
where ai and bi are calculated using RLS algorithm and the
15
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 24, NO. 2, 2018 updating formula is given in (50)–(52): g ( k ) P ( k ) I T P ( k )
( I g ( k ) 1
P ( k 1)
1
T
the general control signal for the vehicle can be written as in (56): (50)
,
1 p ˆ uv 1 s uv K v ( a Dt v Gv v ), u 1 u K ( D p Gˆ ). a t 1 s
(51)
) P ( k ), T
( k 1) ( k ) g (k )( x(k 1) (k )).
(52)
Rewriting the rule for fuzzy controller
IV. EXPERIMENTAL RESULTS
~ If x(k ) A i then Gˆ fi k1i xref (k ) k2i x (k ),
The experimental results that have been realised on the 4 WD SSMR have been presented by using AFFOSMC in this section. Also, to show the effectiveness of the proposed controller, the FOSMC is applied for the same speed and direction angle references. In Fig. 4, the square wave speed and trapezoidal direction angle references have been used. The square wave reference is important to test the performance of the controllers in terms of sudden changes. The obtained results show that, the AFFOSMC has better performance than the FOSMC when the sudden changes occurred. Moreover, the AFFOSMC has less tracking error while tracking the reference direction angle. Also, the proposed controller has given faster response to the both reference signal to drive the vehicle on the desired references. In Fig. 5, the experimental results of square wave speed and trapezoidal direction angle references under load of 20 kg are given to investigate the performance of the proposed controller. Comparing graph of both controllers, it is clearly seen that AFFOSMC has better rise time, less tracking error and overshoot and adapts itself better to changing condition than the FOSMC under square wave speed reference. For trapezoidal direction angle reference, although the FOSMC has less overshoot/undershoot while starting to track the step part of the reference, the AFFOSMC is able to decrease the amount of deviations and has less small direction angle error magnitude as seen from the figures.
(53)
where G f i (k ) is the output of the i. fuzzy controller. For this expression, the output of the fuzzy controller can be adjusted as given below R
G f ( k 1)
G fi (k 1) i
i 1
R
.
(54)
i
i 1
As
an
assumption
Gˆ f (k ) G f (k )
for
designing
AFFOSMC,
Gˆ f i (k ) G f i (k ) . Here,
and
G f i (k )
represents the i. component of the system. In case of the system is hardly effected by another rule and while working according to the i. rule, x ( k ) x i ( k ) can be written. As a result, from the assumptions, (55) can be written
Gˆ f i (k ) G fi (k 1) ai G fi (k ) bi k1i xref (k ) k2i xi (k ) , where
k 2i
a i 0 .1 bi
and
k 1i
(55)
0.1 ai bi k 2 i bi
.
These
equations imply that, for every i, a i and bi can be found and thus, k 1i and k 2i can be updated. With this adaptation rule,
0
0.1 15.8
17
18
19
Reference FOSMC AFFOSMC
-0.2 -0.4 0
5
10
15
Error (m/s)
20
Time (sec)
25
30
15
1.4 4.5
6.5
5
10
8.5
15
20
Time (sec)
25
30
35
20
Time (sec)
25
35
0.1 0 -0.1 -0.2 0
FOSMC AFFOSMC
5
10
15
20
Time (sec)
25
(c) (d) Fig. 4. 4TT Experimental results of AFFOSMC and FOSMC for the square speed and trapezoidal direction angle references.
16
30
(b)
0.2
0
10
1.7
-2 0
35
FOSMC AFFOSMC
5
0 -1
(a)
0.5
-0.5 0
Reference FOSMC AFFOSMC
1
Error (rad)
(m/s)
0.2
2
0.2
(rad)
0.4
(56)
30
35
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 24, NO. 2, 2018
0
0.1 15.8
17
18
19
Reference FOSMC AFFOSMC
-0.2 -0.4 0
5
10
Error (m/s)
0.5
15
20
Time (sec) (a)
25
30
10
15
20
Time (sec) (c)
25
30
FOSMC AFFOSMC 1.7
1.4 4.5
-2 0
35
6.5
5
10
0.2
FOSMC AFFOSMC
5
0 -1
0
-0.5 0
Reference
1 (rad)
(m/s)
0.2
2
0.2
Error (rad)
0.4
20
Time (sec) (b)
25
30
35
0.1 0 -0.1 -0.2 0
35
8.5
15
FOSMC AFFOSMC
5
10
15
20
Time (sec) (d)
25
30
35
Fig. 5. 4TT Experimental results of AFFOSMC and FOSMC for the square speed and trapezoidal direction angle references under total load of 20 kg.
V. CONCLUSIONS
[7]
In this paper, the AFFOSMC has been proposed, designed and validated on the 4 WD SSMR for square speed and trapezoidal direction angle references and also under extra load. To compare the proposed controller, the FOSMC is also applied for the same references and extra load condition. The experimental results indicate that the AFFOSMC has better rise time with smaller steady state error, less overshoot and adapt itself to changing condition when compared with the FOSMC. To conclude, the AFFOSMC adapts itself to the changing condition quickly and works very well.
[8]
[9]
[10]
[11]
REFERENCES [1]
[2]
[3]
[4]
[5] [6]
J. Huang, C. Wen, W. Wang, Z.-P. Jiang, “Adaptive output feedback tracking control of a nonholonomic mobile robot”, Automatica, vol. 50, no. 3, pp. 821–831, 2014. DOI: 10.1016/j.automatica.2013.12.036. O. Elshazly, A. Abo-Ismail, H. S. Abbas, Z. Zyada, “Skid steering mobile robot modeling and control”, in Proc. UKACC Int. Conf. on Control, Loughborough, 2014, pp. 62–67. DOI: 10.1109/CONTROL.2014.6915116. C. Yu, X Chen, “Trajectory tracking of wheeled mobile robot by adopting iterative learning control with predictive, current, and past learning items”, Robotica, vol. 33, no. 7, pp. 1393–1414, 2014. DOI: 10.1017/S0263574714000605. K. Orman, A. Basci, A. Derdiyok, “Speed and direction angle control of four wheel drive skid-steered mobile robot by using fractional order PI controller”, Elektronika ir Elektrotechnika, vol. 22, no. 5, pp. 14–19, 2016. DOI: 10.5755/j01.eie.22.5.16337. L. Xin, Q. Wang, J. She, Y. Li, “Robust adaptive tracking control of wheeled mobile robot”, Robotics and Autonomous Systems, vol. 78, pp. 36–48, 2016 DOI: 10.1016/j.robot. 2016.01.002. Y. Tang, X. Zhang, D. Zhang, G. Zhao, X. Guan, “Fractional order sliding mode controller design for antilock braking systems”, Neurocomputing, vol. 111, pp. 122–130, 2013. DOI:
[12]
[13] [14]
[15]
[16] [17]
17
10.1016/j.neucom.2012.12.019. A. Saghafinia, H. W. Ping, M. N. Uddin, K. S. Gaeid, “Adaptive fuzzy sliding-mode control into chattering-free IM drive”, IEEE Trans. Industry Applications, vol. 51, no. 1, pp. 692–701, 2015. DOI: 10.1109/TIA.2014.2328711. A. Saghafinia, H. W. Ping, M. N. Uddin, and K. S. Gaeid, “Adaptive fuzzy sliding-mode control into chattering-free IM drive”, IEEE Transactions on Industry Applications, vol. 51(1), pp. 692-701, 2015. DOI: 10.1109/TIA.2014.2328711. H. Li, J. Wang, and P. Shi, “Output-feedback based sliding mode control for fuzzy systems with actuator saturation”, IEEE Transactions on Fuzzy Systems, vol. 24(6), pp. 1282-1293, 2016. DOI: 10.1109/TFUZZ.2015.2513085. M. Begnini, D. W. Bertol, N. A. Martins, “A robust adaptive fuzzy variable structure tracking control for the wheeled mobile robot: Simulation and experimental results”, Control Engineering Practice, vol. 64, pp. 27-43, 2017. DOI: 10.1016/j.conengprac.2017.04.006. I. Salgado, D. Cruz-Ortiz, O. Camacho and I. Chairez, “Output feedback control of a skid-steered mobile robot based on the supertwisting algorithm”, Control Engineering Practice, vol. 58, pp. 193203, 2017. DOI: 10.1016/j.conengprac.2016.10.003. M. Ertugrul, A. Sabanoviç and O. Kaynak, 1994. “Various VSS Techniques on The Control of Automated GuidedVehicles and Autonomous Mobile Robots”. PROJECT Report of CAD/CAM Robotics Dept. TUBITAK Marmara Research Center, Gebze-Kocaeli, Turkey. K.B. Oldham and J. Spainer, The Fractional Calculus. Academic Press, 1974. A. Oustaloup, F. Levron, B. Mathieu and F. M. Nanot, “Frequency Band Complex Noninteger Differentiator: Characterization and Synthesis”, IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 1, pp. 25-39, 2000. DOI: 10.1109/81.817385. A. Basci and A. Derdiyok, “Application of Chattering-Free Sliding Mode Controller in Coupled Tank Liquid Level System”, Korean J. Chem. Eng., vol. 30(3), pp. 540-545, 2013. DOI: 10.1007/s11814012-0177-y. A. Sabanovic, K. Jezernik and K. Wada, “Chattering-free sliding modes in robotic manipulators control”, Robotica, vol. 14(1), pp.17– 29, 1996. DOI: 10.1017/S0263574700018907. V. Utkin, G. Jürgen, and S. Jingxin. Sliding mode control in electromechanical systems. CRC press, 2017.
