Abstract—The research work presented within this paper deals with an innovative second-order sliding mode control (SOSMC) allocated to adaptive gain and associated with nonlinear systems subject to unknown but bounded uncertainties. The derived controller guarantees the control gain dynamical adaptation for the sake of counteracting the system’s uncertainties and to mitigate the chattering phenomenon. The Lyapunov method is also used to analyses the stability of any closed loop system (CLS) within a finite- time under bounded uncertainties assumptions. To assess how effective is the approach considered within this paper, the adaptive controller has been carefully studied on a benchmark of nonlinear systems on a damped overturned pendulum.
Index Terms—Control, Gain, Finite-Time, Sliding-Mode, Stability, Adaptive, Nonlinear System, Uncertainty.
I. INTRODUCTION
A damped overturned pendulum cart is an interesting non-linear system, which is an under-actuated mechanical system including several less number of inputs compared to the degrees of freedom [1]. The system was extensively used as a benchmark for testing various control algorithms due to its highly unstable nonlinear dynamics. controlling a nonlinear system happens when the cart is moved along the horizontal direction in between two positions while the pendulum is kept at the upright position known to have the lowest level of mechanical oscillations. from. In fact, the essential shortcoming is to pivot the pendulum from the downward vertical position and keeping the cart in a stable state. However, several control strategies were considered for the sake of better stabilizing and balancing the entire pendulum system, such as energy control [2], event-based control method [3], adaptive control [4], and sliding mode control [5,6,7].
Recently, sliding mode controllers (SMC) for dynamic systems under significant uncertainties has received considerable attention [8,9,10,11,12]. That was thanks to the potential of the SMC to converge in finite-time following an expected trajectory toward an equilibrium state. Moreover, SMC systems can exhibit closed-loop insensitivity to system uncertainties. Indeed, to design a SMC, a sliding surface must be defined where the system’s states can slide till they find their anticipated destinations.
Then, coming-up with a control law in order to push any
Published on December 5, 2019
Authors are with the Electrical Engineering Department, Kairouan University, High Institute of Applied Sciences and Technology, Kairouan, Tunisia.
Corresponding author: Tarek Selmi (E-mail: [email protected])
state outside the sliding surface to the sliding manifold in timely manner must take place [13]. The finite- time stability of the sliding variables was verified using the results of the Lyapunov theory. In [14], it was reported that stable and finite-time systems are more robust and exhibit flexible disturbance rejection properties if a faster convergence. Speaking of the finite time Lyapunov stability described in [15], an equilibrium state in finite time could always meet [16].
On the other hand, the essential shortcoming of classical SMC is what is very often called chattering problem induced by the discontinuous term of the controller.
Nevertheless, the chattering phenomenon could fatigue the system actuators and even dam- age its mechanical parts [17,18]. To solve the above challenge, [19] claims that the controller should use a continuous function instead of a discontinues one. Differently from the boundary layer technique, the so-called high-order sliding mode control (HOSMC) methodology in which the time derivative of the control and its derivatives appear explicitly can be applied to avoid the chatter [20,21]. In fact, the principle of the HOSMC is to compensate the chattering caused by the discontinuity by driving the high-order derivatives of sliding variables to the sliding manifold. The 2nd order SMC, (SOSMC), methodology introduced at its early stage in [22]
could be conservative in terms of robustness and system performances from one hand, and to mitigate the chattering phenomenon from the other hand. Various methods of SOSMC including the conventional twisting algorithms have been widely studied recently in [23,24,25] and applied to some real systems, such as mars entry vehicles [26], induction motors [27] and wind energy [28]. Meanwhile, using adaptive sliding mode control (ASMC) methods is another way to decrease the chattering effect [29,30,31,32].
In this paper, a novel adaptive SOSMC would be synthetized to handle a nonlinear system in the presence of a class of uncertainty. The proposed approach does not require the knowledge of the control gain bounds and it allows the reduction of the chattering phenomenon as well as the convergence of the sliding variables to the origin in infinite- time. In addition, it allows in adjusting the tuning parameters of the controller whilst guaranteeing a sliding motion.
The main contribution of the paper is threefold. First, the proposed controller can avoid the chattering phenomenon and to reduce the control effort. Second, the system, operating under closed-loop considerations, becomes stable system and converges in finite-time to the anticipated system states.
Finally, it contributes a controller handling the control
Adaptive and Robust Second Order Sliding Mode Controller Dedicated to Nonlinear Uncertain Pendulum
Application System
Hedi Dhouibi, Jalel Ghabi, and Tarek Selmi
Vol. 4, No. 12, December 2019
problem of a damped inverted pendulum cart. The proposed controller demonstrated robust behavior against system uncertainties, high tracking precision, fast response and insensitivity of the closed-loop system (CLS).
This paper is subdivided such that section II focuses on nonlinear dynamical modeling of uncertain systems and introduces some useful assumptions. The fundamental results of the controller are presented in section III. The dynamical model of the damped pendulum system is presented within section IV, whereas section V provides the simulated results to demonstrate the utility of the control design. Finally, conclusion and future recommendations are given in section VI.
II. NONLINEAR MODEL STRUCTURE Let us consider the model given by expression (1):
( ) ( , ) ( , )
+( ( , ) ( , )) ( ) ( ) x t f x t f x t
g x t g x t u t w t
(1)Where
x
is the state vector (x X n), u is the input control to be implemented and
( )t nis the external disturbance vector. The vectors( ) n and g( ) n
f are smooth known functions for all
x X . f (.) and (.) g
represent respectively the parametric and control gain uncertainties f( ) and ( ) g .From (1), a generalized uncertainty d x t( , ) ncan be constructed as:
( , ) ( , ) ( , ) ( ) ( )
d x t f x t g x t u t w t
(2) For instance, most matching conditions seem to be not well satisfied [32].Therefore, equation (1) would be expressed differently as in (3):
( , ) ( , ) ( ) ( , )
x f x t g x t u t d x t
(3) To come-up with a widespread analysis of the system previously presented, two suppositions were considered within this paper; which are presented next:i. The continuous control input, u, is an affine scalar function, called Lebesgue-measurable signal and it is such that it satisfies the inequality presented next:
u t ( ) u
max
(4) ii. The function d x t( , ) that represents the generalized uncertainty is delimited by L2 and fulfils the inequality presented next:[35]:
d x t ( , ) d
(5)Where the operator
.
represents the second norm of a given matrix or vector andd
is a positive scalar.III. MAIN RESULTS
In this work, we focus on designing an adaptive SOSMC for expression (3) such that the CLS is considered robust against the uncertainty satisfying the condition (5). Upon this hypothesis, there is no doubts that the CLS reaches the sliding surface within finite-time.
A. Preliminaries
This subsection states according to the literature results of the conventional SMC needed for the main controller.
For the non-linear uncertain system (3),) let us consider:
1
( , ) ( ( ) ( )) ( ( ) ( ))
n
d i i id
i
s x t C x t x t c x t x t
(6)Where C c c1, 2,...,cnT 1nwith
cn1
represents a well-defined constant matrix such that the product ,
Cg x t
is not equal to zero; furthermore, the system, operating under closed loop conditions, would show the expected behavior.( ) ( )
d( )
e t x t x t
is the vector that tracks the system’s error and x td( ) x1d( ),...,t xnd( )t T nis the anticipated state vector.Substitute the state equation from (3) into the first derivative of
s x t ( , )
leads to:( , ) ( , ) ( , ) ( ) ( , ) ( )
( , ) ( , ) ( ) ( , )
s x t Cf x t Cg x t u t Cd x t Cx td
F x t G x t u t
x t
(7)
With
G( )
and( ) F
both
and are presented next:
( , ) ( , ), ( , ) ( , )
d( ) G x t Cg x t F x t C f x t x t
(8)The equivalent uncertainty
( ) t
is given by:
( , ) x t Cd x t ( , ) C f x t u t ( , ) ( ) w t ( )
(9)To make it simple we use
( ) t
instead of ( , ) x t
. The following assumptions will be considered within what follows.The functions
F ( ) and G( )
fulfils the inequality shown next:M
,
m MF F G G G
(10) Where FM, Gm and GM are positive scalars
The so-called equivalent uncertainty
( ) t
is delimited such that ∀x X
:( ) t
(11)where
is a known positive constant.In sliding mode dynamics, the control input, u, is such that it satisfies the following condition [17, 34]:
1
2( , ) ( , ) ( , ) ( , ) 2
d s x t s x t s x t s x t
dt
(12)Where
is a constant positive. Condition (12) can then be written as:
s x t sign s x t ( , ) ( ( , ))
(13)where sign(s) is the sign function defined as:
1, s 0 ( ) ( ( , )) 0, s = 0 1, s < 0
if sign s sign s x t if
if
(14)
Considering (7) and (13), it follows that:
F x t ( , ) G x t u t ( , ) ( ) ( ) t sign s x t ( ( , ))
(15) In conventional SMC, the sliding mode control law is written as the sum of two terms: The equivalent control law responsible for the nominal system performance and used to drive the sliding variables to the sliding surface.
The discontinuous control term that rejects the disturbance of the system [34].
Thus, the control input satisfying condition (15) is given by:
( )
1( , ) ( , ) ( ( , ))
u t G
x t F x t sign s x t
(16) where 0
is the control gain.Let us put into account the below theorem.
Theorem 1. For the nonlinear uncertain system (3) with control law (16) and sliding mode dynamics (7), the CLS is asymptotically stable.
Proof. In the concept of sliding mode control, the following candidate Lyapunov function satisfies Lyapunov stability criteria:
1 2
( , ) ( , )
V x t 2s x t (17) Differentiating Lyapunov function in (17) yields:
( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( ) ( ) V x t s x t s x t
s x t F x t G x t u t t
(18)If the control input from 16 is substituted by 18, the result is:
( , ) ( , ) ( ) ( )
( , ) ( ) ( )
V x t s x t sign s t s x t t sign s
(19)Note that the negative neatness of
V
would no matter be satisfied, as long as the equivalent uncertainty ( ) t
fulfilsthe condition presented next:
( ) t min( , ) t 0
(20) Because the derivative of the Lyapunov function is negative definite which guarantees the motion of the state trajectory to the manifold, the control law presented in (16) ensures the stability and the robustness of the closed loop nonlinear uncertain system given by expression (3). Hence, theorem 1 is demonstrated.Remark 1. The essential shortcoming of the classical SMC is certainly the chattering phenomena. This is due to the discontinuity term, namely,
sign s ( )
. To get rid of a such phenomenon, a saturation function can be employed instead of the sign function in the control law (16) at the price of losing some tracking precision and disturbance rejection properties.In this case, a stable sliding mode control law could be easily deduced as follows:
( )
1( , ) ( , ) ( ( , ))
u t G
x t F x t sat s x t
(21) Wheresat
s represents the saturation function and is given as follows in (22):
( ), ( ) ,
sign s if s sat s s
if s
(22)
where
stands for the boundary thickness layer [35].B. Design of Adaptive Model Control
Consider the new sliding function _ Ë R defined according to variables s and
s
by:
( , ) x t s x t ( , ) s x t ( , )
(23)where
is a positive constant.Notice that in real application, it is only necessary to consider the real sliding mode.
Definition 1. As per [36], the effective sliding mode corresponds to the behavior of the expression (3) when
0
such that: ( , )
S x X x t
(24) where S is assumed locally an integral set in the Flip sense [15].Differentiating the sliding function,
( , ) x t
, one time leads to:
( , ) x t s x t ( , ) s x t ( , )
(25) Substituting the equation (3) into (25) leads:
( , ) x t F x t ( , ) G x t u t ( , ) ( ) ( ) t s x t ( , )
(26)It follows from (26) that the control input with adaptive
Vol. 4, No. 12, December 2019
gain is given by:
1
( , ) ( , ) ( ) ( ( , ))
( ) ( ( , )) F x t s x t
u t G x t
K t sign x t
(27)where K(t) is the adjustable gain expressed next:
if
( , ) x t , ( ) K t
represents a root of:
K t ( ) k
1 ( , ) x t
(28) such thatK (0) k
0 0, k
1 0 and 0
when ( , )x t , ( )K t
is given by:
K t ( ) k
2 (29)* *
2 1 1
with 0<k k , k K t ( ) and t
denotes the first instant time the sliding mode begins.Based on equations (27) through (29), the working principle of the adaptive controller is summarized next:
If
( , ) x t
,the control gain, K(t), is raising up and that is based essentially on equation (28). such raise-up keeps going on until the establishment of the sliding mode at a given time, t1, hence, equation (29) is then considered. In this case, the adaptive controller allows the finite-time stabilization of the sliding variable
by keeping the gain K(t) at the smallest level.The next theorem gives an enough condition under which the sliding motion is stable.
Theorem 2. Considering the system (3) with the sliding variable dynamics (26) and under Assumptions 3-4, the controller (27)-(29) can drive the states of the CLS to the sliding surface (23) in a finite-time.
Proof of Theorem 2. Choosing the Lyapunov function as follows:
1 2 1 2
( , ) ( ( ) )
2 2
V
x t K t k (30) Such that t 0, ( ) K t k
The differentiation of equation (30) as function of the time leads to:
( )
1 ( )
V K k K
F Gu s K k K
(31)
recalling for equation (27) and substituting it within equation (30) leads to:
( )
( )( )
V Ksign K k K
K K k K
(32)
Case
( , )x t
, equation (32) leads to:
1
1 1
1 0 1
0
( )
( )
( )
( )
V K K k k
Kk K sign kk
kk k sign kk
k sign
(33)
Note that if the
( ) t
fulfils (34), thenV
, expressed in (35), will be satisfied.
0
2
(34) and
(35)
( ) min , , 0
( ) min , , 0
t k t
t k t
2
(36)( )
( )
V K K k K
k sign
IV. CART INVERTED PENDULUM SYSTEM DYNAMIC MODEL As shown in Fig.1, by having two degrees of freedom, the damped pendulum could easily be represented by means of two generalized coordinates: the displacement and the deviation, x and
respectively.The control task consists of applying a normalized force f to the cart, which is also the input of the system, to move it forward and backward, and further balancing the associated pendulum from its stable equilibrium point at the upright position.
Table I contains the parameters of the damped inverted pendulum cart used for simulation.
The differential equations presented next, which are based on the Euler Lagrange method, describe the dynamical behavior model of the damped inverted pendulum cart [37]:
1 x cos
2sin
1x f
(37)cos sin
20
x
(38) where 0
is a constant scalar which depends on both the cart and pendulum masses.
1x
and
2 are the viscous frictions of the cart and the pendulum respectively which are considered as linear functions of the cart and the pendulum velocities.Fig. 1. A representative figure of the inverted pendulum cart system.
TABLEI:PARAMETERS OF THE INVERTED PENDULUM CART MODEL
Symbol Definition Value Unit
m Mass of the pendulum 0.25 kg
M Mass of the chart 4 kg
l Length of the pendulum 0.35 m
g Gravitational constant 9.8 m/s2
λ1 Viscous friction coefficient of the cart 0.2 - λ2 Viscous friction coefficient of the pendulum 0.45 -
It should be noted that the physical parameters
,
1and
2 are given by [38]:1 2
1 1/2 3/2 2 1/2 3/2
, ,
M
m ml g ml g
where
1and 2
are the dissipation coefficients presented in the non-actuated coordinate
and the actuated coordinate x respectively.By rewriting equations (37)-(38), equations (39) and (40) could be obtained:
2
2 12
sin cos sin
sin
x f
x
(39)
2 2
2 1
2
sin 1 cos
sin
sin cos
sin f
x
(40)
Let us consider the following state variables:
1 1
,
2,
3,
4x x x x x x
Upon solving equations (39) through (40), a state space model that describes the damped cart-inverted pendulum describing motion could be written:
1 2
2
2 4 3 3 4 3 1 2
2 2
3
3 4
3 2 4
4 2
3 2
1 2 4 3 3 3
2 3
sin cos sin
sin
sin 1
sin
sin cos cos
sin
x x
x x x x x x u
x x
x x
x x
x x
x x x x u x
x
(41)
V. SIMULATION RESULTS
The effectiveness of the anticipated adaptive sliding mode controller is carried out while considering the damped inverted pendulum cart described through the nonlinear model given in (41).
As desired states, we take
x
1d( ) t 2
as the cart position andx
3d( ) t 0
as the angular deviation.starting from the fact that
1
(0) 0.25, (0)
10, (0)
30.1 (0)
30
x x x and x
are respectively the cart position, the cart velocity, the angular deviation and the angular velocity, the system robustness could be validated by simply selecting
0.15
all along with the coefficients of the matrix C chosen as follows: c1 = 0.25, c2 = 0.70, c3 = 1.15, c4 = 1To this end, the tuning parameters of the controller (27)- (29) were chosen as:
k0 = 0.95, k1 = 2.5, k2 = 1.25,
1.5, 0.1
The simulation results are shown in Fig. 2 through Fig. 4 such that Fig. 2 shows the evolution of the system states while Fig. 3 illustrates the control signal u(t) and the adjustable control gain K.t, respectively. Fig. 4 depicts the sliding function
( , ) x t
and the phase trajectory
( , ), ( , )x t
x t
.Fig. 2: Evolution of the system’s states (solid lines) and desired trajectories (dashed lines). (a) Cart displacement. (b) angular deviation.
Fig. 3. Evolution of the control signal and the adaptive control gain. (a) Control signal u. (b) Adaptive control gain K
From Fig. 2, as expected, the proposed controller helped to reach the anticipated position of the cart as well as the expected deviation of the pendulum in a timely manner
6
t and
t 8
, time units respectively). to their desired final values. Therefore, the system’s states follow perfectly their anticipated paths. Nevertheless, the controller quickly drives the tracking errors to zero, which exhibits the finite- time stability of the resultant CLS. However, it can be seen from Fig. 3.a and Fig. 4.a that the proposed controller could moderate the chattering phenomenon. Besides, Fig. 3.bVol. 4, No. 12, December 2019
shows how effective is the amendable gain K t which ( ) compensates the uncertainties of system.
Fig. 4. Evolution of the sliding function and the phase trajectory. (a) Sliding function
. (b) Phase trajectory
,
To make sure that the system is robust under closed loop consideration, a scenario where an external disturbance
1( ) 0.2
d t affects the cart displacement x t at instant time 1( ) 15
t is considered.
The obtained result is shown in Fig. 5 through Fig. 6.
It can be clearly seen from Fig. 5 that under the controller (27)-(29), the states perturbed at t15 recovered their desired states at about t18 time unit later. Nevertheless, this controller effectively compensates the disturbance (see Fig. 6).
It has the potential to conduct the system state trajectories towards the sliding manifold and to maintain it thereafter.
The present controller not only demonstrates robust behavior against parameter variations compared to a first order, but it also quickly removes the uncertainties in the model and pilots the dynamics to nominal values.
VI. CONCLUSION
Within this paper a novel approach to a SOSMC having gain adaptation to handle a nonlinear system subjects to unknown but bounded uncertainty, is presented
Thanks to the Lyapunov approach, of the CLS has been verified in terms of the finite-time stability. The simulations results conducted through the paper valid the effectiveness and simplicity of the proposed design by controlling a damped inverted pendulum cart system. The future work shall focus on extending the result to the adaptive higher order SMC for MIMO nonlinear uncertain systems.
Fig. 5. Evolution of the system’s states (solid lines) and desired trajectories (dashed lines) for perturbed CLS. (a) Cart displacement. (b) Angular
deviation.
Fig. 6. Evolution of the control signal and the adaptive control gain for perturbed CLS. (a) Control signal u. (b) Adaptive control gain K.
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Hedi Dhouibi received his PhD degree in Automatic Control from the Institute of Sciences and Technologies of Lille, France in 2005. He is currently a professor at University of Kairouan and director of the Higher Institute of Applied Sciences and Technology of Kairouan. His research interests include intelligent control, Sliding mode Control, Diagnosis and fault detection.
Jalel Ghabi received his Ph.D. degree in Automatic Control from The National Engineering School of Monastir, University of Monastir, Tunisia in 2009.
He is currently an associate professor at Higher Institute of Applied Sciences and Technology of Kairouan, University of Kairouan,Tunisia. His research interests include Sliding mode control, robust predictive control and Diagnosis.
Tarek Selmi received the B.Sc. degree from Tunis University of Sciences in 2002, the M.Sc. degree from Monastir University of Sciences in 2007 and the PhD degree from the National Engineering School of Sfax, Tunisia in 2013. He started his academic career as high school teacher (2000-2008), then instructor at the vocational college of technology, Saham, Oman (2008-2013). He worked also as Assistant professor of electrical engineering at the Australian college of Kuwait for three years and served as deputy head of the electrical engineering department there (2013-2016). Currently, he works as assistant professor of mechatronics at the Faculty of Engineering, Sohar University, Oman. His research interests include Microcontrollers based systems, Mechatronic systems, and Programmable logic controllers.
